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Nodes and weights of Gauss–Hermite quadrature
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Numbers
$n$
$k$
$x_k$ or $w_k$
1
1
$x_k$:
0
equals: Zero
comment: $x_1=0$, the one-point rule
1
1
$w_k$:
1.772453850905516027298167483341145182797549456122387128213807789852911284591032181374950656738544665
comment: $w_1=\sqrt{\pi}=\Gamma(1/2)$, the integral of $e^{-x^2}$
equals: Values_of_the_Gamma_function_at_rational_numbers#1/2
2
1
$x_k$:
-0.7071067811865475244008443621048490392848359376884740365883398689953662392310535194251937671638207864
comment: $x_1=-1/\sqrt{2}$
equals: Algebraic_numbers_of_degree_2#2,0,-1,1
2
1
$w_k$:
0.8862269254527580136490837416705725913987747280611935641069038949264556422955160906874753283692723327
comment: $w_1=\sqrt{\pi}/2=\Gamma(3/2)$
equals: Values_of_the_Gamma_function_at_rational_numbers#3/2
2
2
$x_k$:
0.7071067811865475244008443621048490392848359376884740365883398689953662392310535194251937671638207864
comment: $x_2=1/\sqrt{2}$
equals: Algebraic_numbers_of_degree_2#2,0,-1,2
2
2
$w_k$:
0.8862269254527580136490837416705725913987747280611935641069038949264556422955160906874753283692723327
comment: $w_2=\sqrt{\pi}/2=\Gamma(3/2)$
equals: Values_of_the_Gamma_function_at_rational_numbers#3/2
3
1
$x_k$:
-1.224744871391589049098642037352945695982973740328335064216346283625480188728657513269929716552320117
comment: $x_1=-\sqrt{3/2}$
equals: Algebraic_numbers_of_degree_2#2,0,-3,1
3
1
$w_k$:
0.2954089751509193378830279138901908637995915760203978547023012983088185474318386968958251094564241109
comment: $w_1=\sqrt{\pi}/6$
3
2
$x_k$:
0
equals: Zero
comment: $x_{2}=0$, the central node of every rule of odd order
3
2
$w_k$:
1.181635900603677351532111655560763455198366304081591418809205193235274189727354787583300437825696444
comment: $w_{2}=\tfrac{2}{3}\sqrt{\pi}$
3
3
$x_k$:
1.224744871391589049098642037352945695982973740328335064216346283625480188728657513269929716552320117
comment: $x_3=\sqrt{3/2}$
equals: Algebraic_numbers_of_degree_2#2,0,-3,2
3
3
$w_k$:
0.2954089751509193378830279138901908637995915760203978547023012983088185474318386968958251094564241109
comment: $w_3=\sqrt{\pi}/6$
4
1
$x_k$:
-1.650680123885784555883341111120745543789434706903775707030373323342057798778707817302550489035530333
comment: $x_1=-\sqrt{\bigl(3+\sqrt{6}\bigr)/2}$
4
1
$w_k$:
0.08131283544724517714303455718988841176333291663832381856534578357804371850461018105201285673391174281
comment: $w_1=\sqrt{\pi}\,(3-\sqrt{6})/12$
4
2
$x_k$:
-0.5246476232752903178840602538347413414135785651694633719018601754437852126251738236361455531722503710
comment: $x_2=-\sqrt{\bigl(3-\sqrt{6}\bigr)/2}$
4
2
$w_k$:
0.8049140900055128365060491844806841796354418114228697455415581113484119237909059096354624716353605899
comment: $w_2=\sqrt{\pi}\,(3+\sqrt{6})/12$
4
3
$x_k$:
0.5246476232752903178840602538347413414135785651694633719018601754437852126251738236361455531722503710
comment: $x_3=\sqrt{\bigl(3-\sqrt{6}\bigr)/2}$
4
3
$w_k$:
0.8049140900055128365060491844806841796354418114228697455415581113484119237909059096354624716353605899
comment: $w_3=\sqrt{\pi}\,(3+\sqrt{6})/12$
4
4
$x_k$:
1.650680123885784555883341111120745543789434706903775707030373323342057798778707817302550489035530333
comment: $x_4=\sqrt{\bigl(3+\sqrt{6}\bigr)/2}$
4
4
$w_k$:
0.08131283544724517714303455718988841176333291663832381856534578357804371850461018105201285673391174281
comment: $w_4=\sqrt{\pi}\,(3-\sqrt{6})/12$
5
1
$x_k$:
-2.020182870456085632928724088144645147052232147465047392683263162405991380184033204210731765879926476
comment: $x_1=-\sqrt{\bigl(5+\sqrt{10}\bigr)/2}$
5
1
$w_k$:
0.01995324205904591320774345859417357486456997737391905612633246236109687061389082201586476536867736517
comment: $w_1=\sqrt{\pi}\,(7-2\sqrt{10})/60$
5
2
$x_k$:
-0.9585724646138185071127705938929883181860888552581885315861435073024960334291329430216049978274590246
comment: $x_2=-\sqrt{\bigl(5-\sqrt{10}\bigr)/2}$
5
2
$w_k$:
0.3936193231522411598284956208520936344548582290546379404568893552712490957906833536382903878703163901
comment: $w_2=\sqrt{\pi}\,(7+2\sqrt{10})/60$
5
3
$x_k$:
0
equals: Zero
comment: $x_{3}=0$, the central node of every rule of odd order
5
3
$w_k$:
0.9453087204829418812256893244486107641586930432652731350473641545882193517818838300666403502605571549
comment: $w_{3}=\tfrac{8}{15}\sqrt{\pi}$
5
4
$x_k$:
0.9585724646138185071127705938929883181860888552581885315861435073024960334291329430216049978274590246
comment: $x_4=\sqrt{\bigl(5-\sqrt{10}\bigr)/2}$
5
4
$w_k$:
0.3936193231522411598284956208520936344548582290546379404568893552712490957906833536382903878703163901
comment: $w_4=\sqrt{\pi}\,(7+2\sqrt{10})/60$
5
5
$x_k$:
2.020182870456085632928724088144645147052232147465047392683263162405991380184033204210731765879926476
comment: $x_5=\sqrt{\bigl(5+\sqrt{10}\bigr)/2}$
5
5
$w_k$:
0.01995324205904591320774345859417357486456997737391905612633246236109687061389082201586476536867736517
comment: $w_5=\sqrt{\pi}\,(7-2\sqrt{10})/60$
6
1
$x_k$:
-2.350604973674492222833921987060920846537863827294099092297386527143522663733164041215386513198952701
6
1
$w_k$:
0.004530009905508845640857472564627150932510071940439710097674472694996969098162824102407063265172580885
6
2
$x_k$:
-1.335849074013696949714895282970370673972589526125010813994814429198824379497946053728323020745502497
6
2
$w_k$:
0.1570673203228566439163115635083782689123146500357077333048858982128069186653074531204681628864169344
6
3
$x_k$:
-0.4360774119276165086792159482506249107662566689318268058515491611448032171656217290290433116765577597
6
3
$w_k$:
0.7246295952243925240919147055975671715539500060850461207043435240186517545320458134646001022176828175
6
4
$x_k$:
0.4360774119276165086792159482506249107662566689318268058515491611448032171656217290290433116765577597
6
4
$w_k$:
0.7246295952243925240919147055975671715539500060850461207043435240186517545320458134646001022176828175
6
5
$x_k$:
1.335849074013696949714895282970370673972589526125010813994814429198824379497946053728323020745502497
6
5
$w_k$:
0.1570673203228566439163115635083782689123146500357077333048858982128069186653074531204681628864169344
6
6
$x_k$:
2.350604973674492222833921987060920846537863827294099092297386527143522663733164041215386513198952701
6
6
$w_k$:
0.004530009905508845640857472564627150932510071940439710097674472694996969098162824102407063265172580885
7
1
$x_k$:
-2.651961356835233492447082006516616114438158478625529417203100307147160094901603166805290681869787864
7
1
$w_k$:
0.0009717812450995191541494242559389596444424012170142016498000143379833414269858014603119871827105122041
7
2
$x_k$:
-1.673551628767471445031801398303594819107810057735408926924217509993713833369034798661254076179380506
7
2
$w_k$:
0.05451558281912703059217856884169512596089991585934649298658684663136944856271119546554645460303904360
7
3
$x_k$:
-0.8162878828589646630387109590271458167428894003786361568447220334359490704876651166851979497670411667
7
3
$w_k$:
0.4256072526101278005203174666663910354397068210140015258787895391335802729707260237330567364718554248
7
4
$x_k$:
0
equals: Zero
comment: $x_{4}=0$, the central node of every rule of odd order
7
4
$w_k$:
0.8102646175568073267648765638130949407074511799416626871834549896470451586701861400571203002233347042
comment: $w_{4}=\tfrac{16}{35}\sqrt{\pi}$
7
5
$x_k$:
0.8162878828589646630387109590271458167428894003786361568447220334359490704876651166851979497670411667
7
5
$w_k$:
0.4256072526101278005203174666663910354397068210140015258787895391335802729707260237330567364718554248
7
6
$x_k$:
1.673551628767471445031801398303594819107810057735408926924217509993713833369034798661254076179380506
7
6
$w_k$:
0.05451558281912703059217856884169512596089991585934649298658684663136944856271119546554645460303904360
7
7
$x_k$:
2.651961356835233492447082006516616114438158478625529417203100307147160094901603166805290681869787864
7
7
$w_k$:
0.0009717812450995191541494242559389596444424012170142016498000143379833414269858014603119871827105122041
8
1
$x_k$:
-2.930637420257244019223502705243599146199448585521638410171381787565020600694777022801225218528478343
8
1
$w_k$:
0.0001996040722113676192060904525440964562257126380025000994092863971132629439779667896515047668339484132
8
2
$x_k$:
-1.981656756695842925854630639769309568694911634038218595645664931266803097121440396257841291943514498
8
2
$w_k$:
0.01707798300741347545620305643644567818051042485575891588946848278957540593702449435015137358994860051
8
3
$x_k$:
-1.157193712446780194720765779063100243452004759597614018824298435037158025959993751803061204872779138
8
3
$w_k$:
0.2078023258148918795432586202857005575935760268970354867237216853713187630754059062088747610782112574
8
4
$x_k$:
-0.3811869902073221168547188855836914176318600315196565942804019391314297380183854889698688298235784275
8
4
$w_k$:
0.6611470125582412910304159744958822591684625636703966613943044403684482103391077233387976889342785264
8
5
$x_k$:
0.3811869902073221168547188855836914176318600315196565942804019391314297380183854889698688298235784275
8
5
$w_k$:
0.6611470125582412910304159744958822591684625636703966613943044403684482103391077233387976889342785264
8
6
$x_k$:
1.157193712446780194720765779063100243452004759597614018824298435037158025959993751803061204872779138
8
6
$w_k$:
0.2078023258148918795432586202857005575935760268970354867237216853713187630754059062088747610782112574
8
7
$x_k$:
1.981656756695842925854630639769309568694911634038218595645664931266803097121440396257841291943514498
8
7
$w_k$:
0.01707798300741347545620305643644567818051042485575891588946848278957540593702449435015137358994860051
8
8
$x_k$:
2.930637420257244019223502705243599146199448585521638410171381787565020600694777022801225218528478343
8
8
$w_k$:
0.0001996040722113676192060904525440964562257126380025000994092863971132629439779667896515047668339484132
9
1
$x_k$:
-3.190993201781527607230047795380909124968962122238229442611564150162490638257915304530452575357440644
9
1
$w_k$:
0.00003960697726326438190458629464254940873729641818060867681993358786689396318255225192415860390327177305
9
2
$x_k$:
-2.266580584531843111802096932837620348551329881425402802096049345124985227075451177360016942848076061
9
2
$w_k$:
0.004943624275536947217224565977633747902769654421181301695541236104311595983731671666305870214502405989
9
3
$x_k$:
-1.468553289216667931667015739248627982727791035639301097087834675832654729296932225614371375920651302
9
3
$w_k$:
0.08847452739437657328797511474756631407490612927141972405501377496248797774929870804856447545166283975
9
4
$x_k$:
-0.7235510187528375733226398645794120947065192122948753356072493363147507864336405371838491024383077339
9
4
$w_k$:
0.4326515590025557501998121129560211174757166790874507353757711770953246596347759853619606906666106133
9
5
$x_k$:
0
equals: Zero
comment: $x_{5}=0$, the central node of every rule of odd order
9
5
$w_k$:
0.7202352156060509571243347233894177250732899377259223886075155463529290299290543467174402668651864037
comment: $w_{5}=\tfrac{128}{315}\sqrt{\pi}$
9
6
$x_k$:
0.7235510187528375733226398645794120947065192122948753356072493363147507864336405371838491024383077339
9
6
$w_k$:
0.4326515590025557501998121129560211174757166790874507353757711770953246596347759853619606906666106133
9
7
$x_k$:
1.468553289216667931667015739248627982727791035639301097087834675832654729296932225614371375920651302
9
7
$w_k$:
0.08847452739437657328797511474756631407490612927141972405501377496248797774929870804856447545166283975
9
8
$x_k$:
2.266580584531843111802096932837620348551329881425402802096049345124985227075451177360016942848076061
9
8
$w_k$:
0.004943624275536947217224565977633747902769654421181301695541236104311595983731671666305870214502405989
9
9
$x_k$:
3.190993201781527607230047795380909124968962122238229442611564150162490638257915304530452575357440644
9
9
$w_k$:
0.00003960697726326438190458629464254940873729641818060867681993358786689396318255225192415860390327177305
10
1
$x_k$:
-3.436159118837737603326725494319121384840678309390177290687176294400236114564951115346862723161884329
10
1
$w_k$:
0.000007640432855232620629159367859595222108289114735832363195288555128091163348000578369089318599319584542
10
2
$x_k$:
-2.532731674232789796408960797754793480307846508156724945933287413493085938047386534022729232594177165
10
2
$w_k$:
0.001343645746781232692201565585845913869869273278625770364616249802076248381841315520178215223437599278
10
3
$x_k$:
-1.756683649299881773451401220106156763295474493738847100087284956096170579609819077127826671146983831
10
3
$w_k$:
0.03387439445548106313616473127758597369816194248186701673971515769977138127781357340291440158444986923
10
4
$x_k$:
-1.036610829789513654177491916759209016298256110657215724437232687403640418724387899298030602824390073
10
4
$w_k$:
0.2401386110823146864165232950058613953700424939019273989906605614799431523716931435060024114986569652
10
5
$x_k$:
-0.3429013272237046087891650255572580312083026586777306282057710390729201819231210156071352030544324806
10
5
$w_k$:
0.6108626337353257987835649904334197132385927292840375456487166373895367691008200576800112107441285794
10
6
$x_k$:
0.3429013272237046087891650255572580312083026586777306282057710390729201819231210156071352030544324806
10
6
$w_k$:
0.6108626337353257987835649904334197132385927292840375456487166373895367691008200576800112107441285794
10
7
$x_k$:
1.036610829789513654177491916759209016298256110657215724437232687403640418724387899298030602824390073
10
7
$w_k$:
0.2401386110823146864165232950058613953700424939019273989906605614799431523716931435060024114986569652
10
8
$x_k$:
1.756683649299881773451401220106156763295474493738847100087284956096170579609819077127826671146983831
10
8
$w_k$:
0.03387439445548106313616473127758597369816194248186701673971515769977138127781357340291440158444986923
10
9
$x_k$:
2.532731674232789796408960797754793480307846508156724945933287413493085938047386534022729232594177165
10
9
$w_k$:
0.001343645746781232692201565585845913869869273278625770364616249802076248381841315520178215223437599278
10
10
$x_k$:
3.436159118837737603326725494319121384840678309390177290687176294400236114564951115346862723161884329
10
10
$w_k$:
0.000007640432855232620629159367859595222108289114735832363195288555128091163348000578369089318599319584542
11
1
$x_k$:
-3.668470846559582518458371464851315149288728928514951377724789675446775309590269179896001972668832982
11
1
$w_k$:
0.000001439560393714258220330883660319426992280608508863842491625095000276373353952235679116908236650768981
11
2
$x_k$:
-2.783290099781651770836718701516373622670806536898230965907314944931969597649723967513507850592162665
11
2
$w_k$:
0.0003468194663233455106434137729401755225442481558159626756293808185825539428148157956960820190783082210
11
3
$x_k$:
-2.025948015825755335165912831212899268507076961252246284210942649551496003832243906458262324171800775
11
3
$w_k$:
0.01191139544491153245038742029158900494487345143354610264190999672094065074966919019543469948211997977
11
4
$x_k$:
-1.326557084494932855949734735582688948469423039875487625771405996678492519345236675897744963495286400
11
4
$w_k$:
0.1172278751677085033817886493084315819712885836690422560451082531701942630262421183410755384855056571
11
5
$x_k$:
-0.6568095668820997650246115753830130641483585458243335092760065461677857463842346593400995347227095043
11
5
$w_k$:
0.4293597523561250284460735986005798163179443743188611113528939362340428845993204019752988610810651897
11
6
$x_k$:
0
equals: Zero
comment: $x_{6}=0$, the central node of every rule of odd order
11
6
$w_k$:
0.6547592869145917792039406576267433864302635797508385350977414057753900272082312242885820607865330943
11
7
$x_k$:
0.6568095668820997650246115753830130641483585458243335092760065461677857463842346593400995347227095043
11
7
$w_k$:
0.4293597523561250284460735986005798163179443743188611113528939362340428845993204019752988610810651897
11
8
$x_k$:
1.326557084494932855949734735582688948469423039875487625771405996678492519345236675897744963495286400
11
8
$w_k$:
0.1172278751677085033817886493084315819712885836690422560451082531701942630262421183410755384855056571
11
9
$x_k$:
2.025948015825755335165912831212899268507076961252246284210942649551496003832243906458262324171800775
11
9
$w_k$:
0.01191139544491153245038742029158900494487345143354610264190999672094065074966919019543469948211997977
11
10
$x_k$:
2.783290099781651770836718701516373622670806536898230965907314944931969597649723967513507850592162665
11
10
$w_k$:
0.0003468194663233455106434137729401755225442481558159626756293808185825539428148157956960820190783082210
11
11
$x_k$:
3.668470846559582518458371464851315149288728928514951377724789675446775309590269179896001972668832982
11
11
$w_k$:
0.000001439560393714258220330883660319426992280608508863842491625095000276373353952235679116908236650768981
12
1
$x_k$:
-3.889724897869781919271642747244191760148181014166481910460268957075634631431076976110772513464122502
12
1
$w_k$:
2.658551684356301606023114008768679442902629821965654578499276306239589462541132379290758005871817398e-7
12
2
$x_k$:
-3.020637025120889771710679375176763646003806598310335307047095859772485405408349785297040061457997515
12
2
$w_k$:
0.00008573687043587858654569063231532409401897054469197432370024358283767882536618251228806965956432173799
12
3
$x_k$:
-2.279507080501059900187728569424341135039156593214033296603212545375105853642387917941930613482576732
12
3
$w_k$:
0.003905390584629061859994384326195310974150004714085757915277837173804248553546796478092495978467797807
12
4
$x_k$:
-1.597682635152604796709662770904576708240286988768717152668888692425366951296233502448726181069297887
12
4
$w_k$:
0.05160798561588392999187344236061309448155103495095191966476447809515001470959463105746801725759760809
12
5
$x_k$:
-0.9477883912401637437045781310601365136791537959145449103039072260782020981153278846158970268878368668
12
5
$w_k$:
0.2604923102641611292333961397652475332448629633579266206196639983338069551449977167948494846036397590
12
6
$x_k$:
-0.3142403762543591112766116340953371289721108383227137090312444801921111994083197916062769928133021921
12
6
$w_k$:
0.5701352362624795783471134822748004517362474642305550950180394878132261211030645097315393317942022589
12
7
$x_k$:
0.3142403762543591112766116340953371289721108383227137090312444801921111994083197916062769928133021921
12
7
$w_k$:
0.5701352362624795783471134822748004517362474642305550950180394878132261211030645097315393317942022589
12
8
$x_k$:
0.9477883912401637437045781310601365136791537959145449103039072260782020981153278846158970268878368668
12
8
$w_k$:
0.2604923102641611292333961397652475332448629633579266206196639983338069551449977167948494846036397590
12
9
$x_k$:
1.597682635152604796709662770904576708240286988768717152668888692425366951296233502448726181069297887
12
9
$w_k$:
0.05160798561588392999187344236061309448155103495095191966476447809515001470959463105746801725759760809
12
10
$x_k$:
2.279507080501059900187728569424341135039156593214033296603212545375105853642387917941930613482576732
12
10
$w_k$:
0.003905390584629061859994384326195310974150004714085757915277837173804248553546796478092495978467797807
12
11
$x_k$:
3.020637025120889771710679375176763646003806598310335307047095859772485405408349785297040061457997515
12
11
$w_k$:
0.00008573687043587858654569063231532409401897054469197432370024358283767882536618251228806965956432173799
12
12
$x_k$:
3.889724897869781919271642747244191760148181014166481910460268957075634631431076976110772513464122502
12
12
$w_k$:
2.658551684356301606023114008768679442902629821965654578499276306239589462541132379290758005871817398e-7
13
1
$x_k$:
-4.101337596178639641178915080071965222414152452616536264318872768360605442273891408308696099367772566
13
1
$w_k$:
4.825731850073131088349973323422195511193364919368248305745734695223467959581997306932751781195216620e-8
13
2
$x_k$:
-3.246608978372409988122051152361008278552958329142365752751629953002364362559883262674750678478830560
13
2
$w_k$:
0.00002043036040270707312486694329367147864683120508067749978671287350479924498187219150999899724579080127
13
3
$x_k$:
-2.519735685678237883430409136282722998566121127996002891441680886201884833232789883496822567548677874
13
3
$w_k$:
0.001207459992719385947309248992236796794538849658245953914840368078765068866763166277543473112512989603
13
4
$x_k$:
-1.853107651601512142003506443164832568536228329064105520406972843556498810669517081470732528362352253
13
4
$w_k$:
0.02086277529616993921660338050500886954469001385546258207293583308253984946728619471267848657520977413
13
5
$x_k$:
-1.220055036590748426222055266373740877302514583227224814854784864189380374752490860281571102594837773
13
5
$w_k$:
0.1403233206870234377627922688730651991913313168207640439878635920909115652899571945662417159524780512
13
6
$x_k$:
-0.6057638791710601130805371086017867227703068406946663140008597789645563437709462131924390064197230728
13
6
$w_k$:
0.4216162968985432217468935585679686418151832600876040967500367594470535484803567328324907101951387946
13
7
$x_k$:
0
equals: Zero
comment: $x_{7}=0$, the central node of every rule of odd order
13
7
$w_k$:
0.6043931879211616423420990685785323567048586890007740323979151437926677174229826685740757484183382409
13
8
$x_k$:
0.6057638791710601130805371086017867227703068406946663140008597789645563437709462131924390064197230728
13
8
$w_k$:
0.4216162968985432217468935585679686418151832600876040967500367594470535484803567328324907101951387946
13
9
$x_k$:
1.220055036590748426222055266373740877302514583227224814854784864189380374752490860281571102594837773
13
9
$w_k$:
0.1403233206870234377627922688730651991913313168207640439878635920909115652899571945662417159524780512
13
10
$x_k$:
1.853107651601512142003506443164832568536228329064105520406972843556498810669517081470732528362352253
13
10
$w_k$:
0.02086277529616993921660338050500886954469001385546258207293583308253984946728619471267848657520977413
13
11
$x_k$:
2.519735685678237883430409136282722998566121127996002891441680886201884833232789883496822567548677874
13
11
$w_k$:
0.001207459992719385947309248992236796794538849658245953914840368078765068866763166277543473112512989603
13
12
$x_k$:
3.246608978372409988122051152361008278552958329142365752751629953002364362559883262674750678478830560
13
12
$w_k$:
0.00002043036040270707312486694329367147864683120508067749978671287350479924498187219150999899724579080127
13
13
$x_k$:
4.101337596178639641178915080071965222414152452616536264318872768360605442273891408308696099367772566
13
13
$w_k$:
4.825731850073131088349973323422195511193364919368248305745734695223467959581997306932751781195216620e-8
14
1
$x_k$:
-4.304448570473631812621298100368942725321810387946283674821897208672823119329674655333184399213805612
14
1
$w_k$:
8.628591168125157945320417834289678809096697086691495719300462015871602627301159752815993609865108354e-9
14
2
$x_k$:
-3.462656933602270550208917361150432442139076537661460591978967383305860628256915679352516273114474947
14
2
$w_k$:
0.000004716484355018916748876889501052372102209361671269098128659835544182487216083180020726558300518241581
14
3
$x_k$:
-2.748470724985402568624998524146430929973795449932230440125820118375273206673692589931851135678809210
14
3
$w_k$:
0.0003550926135519236104836610766906507297836247875284650905487185147738142121199818156889268375258706173
14
4
$x_k$:
-2.095183258507716815734972726303237455479783354448251210298134116562795781209429550051913359535992441
14
4
$w_k$:
0.007850054726457944310486443346075685554189267194778071504289456986676887260095937635354230918551653061
14
5
$x_k$:
-1.476682731141140870583506544205078582665937734637755777952204087217904348952244370433508976406363780
14
5
$w_k$:
0.06850553422346520553871633123670962480736293620750519767459905962286928074262348795825163116311545684
14
6
$x_k$:
-0.8787137873293994161146793118607960726033855750149142661710841447495470623965801148396810956991782850
14
6
$w_k$:
0.2731056090642466033525691870256054663849878675270343712082396314975727458884052166093422514134668394
14
7
$x_k$:
-0.2917455106725620784461130757993811408910884404322091673470162474775011083269960625809071061057020480
14
7
$w_k$:
0.5364059097120901497949212967755722772606700138859791028396026491685567158334527561876578086623183847
14
8
$x_k$:
0.2917455106725620784461130757993811408910884404322091673470162474775011083269960625809071061057020480
14
8
$w_k$:
0.5364059097120901497949212967755722772606700138859791028396026491685567158334527561876578086623183847
14
9
$x_k$:
0.8787137873293994161146793118607960726033855750149142661710841447495470623965801148396810956991782850
14
9
$w_k$:
0.2731056090642466033525691870256054663849878675270343712082396314975727458884052166093422514134668394
14
10
$x_k$:
1.476682731141140870583506544205078582665937734637755777952204087217904348952244370433508976406363780
14
10
$w_k$:
0.06850553422346520553871633123670962480736293620750519767459905962286928074262348795825163116311545684
14
11
$x_k$:
2.095183258507716815734972726303237455479783354448251210298134116562795781209429550051913359535992441
14
11
$w_k$:
0.007850054726457944310486443346075685554189267194778071504289456986676887260095937635354230918551653061
14
12
$x_k$:
2.748470724985402568624998524146430929973795449932230440125820118375273206673692589931851135678809210
14
12
$w_k$:
0.0003550926135519236104836610766906507297836247875284650905487185147738142121199818156889268375258706173
14
13
$x_k$:
3.462656933602270550208917361150432442139076537661460591978967383305860628256915679352516273114474947
14
13
$w_k$:
0.000004716484355018916748876889501052372102209361671269098128659835544182487216083180020726558300518241581
14
14
$x_k$:
4.304448570473631812621298100368942725321810387946283674821897208672823119329674655333184399213805612
14
14
$w_k$:
8.628591168125157945320417834289678809096697086691495719300462015871602627301159752815993609865108354e-9
15
1
$x_k$:
-4.499990707309391553664380530534824219931983053494990593827174284601936622852563058626438955691958996
15
1
$w_k$:
1.522475804253517020160626669648266182796216031601453793001434959953549158269794714037583083893790370e-9
15
2
$x_k$:
-3.669950373404452534729223833115681484695855626010826820082725019771292961232534152513759616527256761
15
2
$w_k$:
0.000001059115547711066635775207910550163287296352401831549915876172845509130326551589368432561816927964130
15
3
$x_k$:
-2.967166927905603248488960363549806315570516407638138150721272752211895691604629404993456164928358729
15
3
$w_k$:
0.0001000044412324998681272967361769785085169938781087638778923410716175215246096650343937891735161400595
15
4
$x_k$:
-2.325732486173857745454044794485347012307095009563482895772162594874023804858360872706184155464310313
15
4
$w_k$:
0.002778068842912775896078870492292134903012425113482266668571706121009897950137220528932794973888018311
15
5
$x_k$:
-1.719992575186488932415831525152561514842919563061388849893821630009513243476183242843971888817612245
15
5
$w_k$:
0.03078003387254608222868141587578014556964507809535783167012620432094342629690748354354310507577256211
15
6
$x_k$:
-1.136115585210920666319134905556102125283529294784665085064839259980071596710100440963196660183048271
15
6
$w_k$:
0.1584889157959357468838393849599940272100270362083442048414741716255649154339707411678421083298628580
15
7
$x_k$:
-0.5650695832555757485260203371981888668155207880050578868093073968274505847091537515997352628303461029
15
7
$w_k$:
0.4120286874988986270258910795678103189337523274169214187784427355097944772086233585523650349549415633
15
8
$x_k$:
0
equals: Zero
comment: $x_{8}=0$, the central node of every rule of odd order
15
8
$w_k$:
0.5641003087264175328526257973399635329245347764007224302380541342064898695947838240024706985237823582
15
9
$x_k$:
0.5650695832555757485260203371981888668155207880050578868093073968274505847091537515997352628303461029
15
9
$w_k$:
0.4120286874988986270258910795678103189337523274169214187784427355097944772086233585523650349549415633
15
10
$x_k$:
1.136115585210920666319134905556102125283529294784665085064839259980071596710100440963196660183048271
15
10
$w_k$:
0.1584889157959357468838393849599940272100270362083442048414741716255649154339707411678421083298628580
15
11
$x_k$:
1.719992575186488932415831525152561514842919563061388849893821630009513243476183242843971888817612245
15
11
$w_k$:
0.03078003387254608222868141587578014556964507809535783167012620432094342629690748354354310507577256211
15
12
$x_k$:
2.325732486173857745454044794485347012307095009563482895772162594874023804858360872706184155464310313
15
12
$w_k$:
0.002778068842912775896078870492292134903012425113482266668571706121009897950137220528932794973888018311
15
13
$x_k$:
2.967166927905603248488960363549806315570516407638138150721272752211895691604629404993456164928358729
15
13
$w_k$:
0.0001000044412324998681272967361769785085169938781087638778923410716175215246096650343937891735161400595
15
14
$x_k$:
3.669950373404452534729223833115681484695855626010826820082725019771292961232534152513759616527256761
15
14
$w_k$:
0.000001059115547711066635775207910550163287296352401831549915876172845509130326551589368432561816927964130
15
15
$x_k$:
4.499990707309391553664380530534824219931983053494990593827174284601936622852563058626438955691958996
15
15
$w_k$:
1.522475804253517020160626669648266182796216031601453793001434959953549158269794714037583083893790370e-9
16
1
$x_k$:
-4.688738939305818364688498648745610890632391727543541815960181362671147768532921842164766368768820765
16
1
$w_k$:
2.654807474011182244709263660502093665621543650307756291118528185266520962462364669132226269530793284e-10
16
2
$x_k$:
-3.869447904860122698719424098014812397006599470249698273241309961573487822989207380232602822351227313
16
2
$w_k$:
2.320980844865210653387494231848608877791018602620217574631636158438535734375033389007888285920115892e-7
16
3
$x_k$:
-3.176999161979956026813994559263696476791667372741911235428063704775521404720186750107164113762624502
16
3
$w_k$:
0.00002711860092537881512018914322435956989760120245689016073075639332698783197944635217453900601227907938
16
4
$x_k$:
-2.546202157847481362159328705445894124519465199405846019896931987850904829462943838207229808992400363
16
4
$w_k$:
0.0009322840086241805299142773055369015268967846050286548140104627082812953051303506030622943522118291159
16
5
$x_k$:
-1.951787990916253977434655414959887492768326058659915935541263213302125411232221344064380773506385555
16
5
$w_k$:
0.01288031153550997368346429993117211463462634628527580270233312531674174779477344956395785445019330637
16
6
$x_k$:
-1.380258539198880796372089669694582038440681769970045641859875916453720401973170033404769524784271352
16
6
$w_k$:
0.08381004139898582941542073490011560472811833580650188882550072715807527531055147783667160674238758103
16
7
$x_k$:
-0.8229514491446558925824544967339426407635081725412670635807022947747622056916678929759594481237216227
16
7
$w_k$:
0.2806474585285336753694633353796526238946703087136865353073367026869640378825765127600608315262888717
16
8
$x_k$:
-0.2734810461381524521582804019650150339298178803279031885167758686448979121775797915283933978604385148
16
8
$w_k$:
0.5079294790166137419135173417905214361334682057842291652444590283875976357902793198219728345901272465
16
9
$x_k$:
0.2734810461381524521582804019650150339298178803279031885167758686448979121775797915283933978604385148
16
9
$w_k$:
0.5079294790166137419135173417905214361334682057842291652444590283875976357902793198219728345901272465
16
10
$x_k$:
0.8229514491446558925824544967339426407635081725412670635807022947747622056916678929759594481237216227
16
10
$w_k$:
0.2806474585285336753694633353796526238946703087136865353073367026869640378825765127600608315262888717
16
11
$x_k$:
1.380258539198880796372089669694582038440681769970045641859875916453720401973170033404769524784271352
16
11
$w_k$:
0.08381004139898582941542073490011560472811833580650188882550072715807527531055147783667160674238758103
16
12
$x_k$:
1.951787990916253977434655414959887492768326058659915935541263213302125411232221344064380773506385555
16
12
$w_k$:
0.01288031153550997368346429993117211463462634628527580270233312531674174779477344956395785445019330637
16
13
$x_k$:
2.546202157847481362159328705445894124519465199405846019896931987850904829462943838207229808992400363
16
13
$w_k$:
0.0009322840086241805299142773055369015268967846050286548140104627082812953051303506030622943522118291159
16
14
$x_k$:
3.176999161979956026813994559263696476791667372741911235428063704775521404720186750107164113762624502
16
14
$w_k$:
0.00002711860092537881512018914322435956989760120245689016073075639332698783197944635217453900601227907938
16
15
$x_k$:
3.869447904860122698719424098014812397006599470249698273241309961573487822989207380232602822351227313
16
15
$w_k$:
2.320980844865210653387494231848608877791018602620217574631636158438535734375033389007888285920115892e-7
16
16
$x_k$:
4.688738939305818364688498648745610890632391727543541815960181362671147768532921842164766368768820765
16
16
$w_k$:
2.654807474011182244709263660502093665621543650307756291118528185266520962462364669132226269530793284e-10
17
1
$x_k$:
-4.871345193674403088349276556620113770860321241460771915159043858232258818564099273842429623481967559
17
1
$w_k$:
4.580578930798633305808892812223474448410255483561813165675248451766693930164575359806371141933901174e-11
17
2
$x_k$:
-4.061946675875474306892455596982681556520053139337483651783628965819263893918267399855646245176898410
17
2
$w_k$:
4.977078981630794052278633537147003058712006013369023777892213640630941397833002710519641247944018371e-8
17
3
$x_k$:
-3.378932091141494083383270692890417243265510291821616973280641171103581900579436172701477991830965412
17
3
$w_k$:
0.000007112289140021309583533273762184088432339250948845026397153022308886708858550827435189772740692449794
17
4
$x_k$:
-2.757762915703888730926403495739532824980936862040772614339117169744522607551057660297031294366063166
17
4
$w_k$:
0.0002986432866977530411513366430594873426081960835460885944499440351080312891731098944849547850627339966
17
5
$x_k$:
-2.173502826666620819275379071487942936193184105173650468834434481976684303447782805377825481546667113
17
5
$w_k$:
0.005067349957627537911700694958787153073187129664315628010739136500467737981059750238213934320875589077
17
6
$x_k$:
-1.612924314221231333112882544536394358363177963976444752974674130497879461117555730603199378461797577
17
6
$w_k$:
0.04092003414975627980949948778538838888682255007952784399557832926047784763431238275649968294557185446
17
7
$x_k$:
-1.067648725743450553630457737989176074609296755460102085809485190954061181203796999244015728761573144
17
7
$w_k$:
0.1726482976700970792176451962187513393563787904038114326343927864431584394425613372379222841457781759
17
8
$x_k$:
-0.5316330013426547313490865537176280941504848574290495673801894865225386520145616876605715735457971260
17
8
$w_k$:
0.4018264694704119565776350852570051353298875550215998936780533954594623403678658896294955066523989274
17
9
$x_k$:
0
equals: Zero
comment: $x_{9}=0$, the central node of every rule of odd order
17
9
$w_k$:
0.5309179376248635603318831033787892074583856719065622872828744792531669360892083049435018339047363371
17
10
$x_k$:
0.5316330013426547313490865537176280941504848574290495673801894865225386520145616876605715735457971260
17
10
$w_k$:
0.4018264694704119565776350852570051353298875550215998936780533954594623403678658896294955066523989274
17
11
$x_k$:
1.067648725743450553630457737989176074609296755460102085809485190954061181203796999244015728761573144
17
11
$w_k$:
0.1726482976700970792176451962187513393563787904038114326343927864431584394425613372379222841457781759
17
12
$x_k$:
1.612924314221231333112882544536394358363177963976444752974674130497879461117555730603199378461797577
17
12
$w_k$:
0.04092003414975627980949948778538838888682255007952784399557832926047784763431238275649968294557185446
17
13
$x_k$:
2.173502826666620819275379071487942936193184105173650468834434481976684303447782805377825481546667113
17
13
$w_k$:
0.005067349957627537911700694958787153073187129664315628010739136500467737981059750238213934320875589077
17
14
$x_k$:
2.757762915703888730926403495739532824980936862040772614339117169744522607551057660297031294366063166
17
14
$w_k$:
0.0002986432866977530411513366430594873426081960835460885944499440351080312891731098944849547850627339966
17
15
$x_k$:
3.378932091141494083383270692890417243265510291821616973280641171103581900579436172701477991830965412
17
15
$w_k$:
0.000007112289140021309583533273762184088432339250948845026397153022308886708858550827435189772740692449794
17
16
$x_k$:
4.061946675875474306892455596982681556520053139337483651783628965819263893918267399855646245176898410
17
16
$w_k$:
4.977078981630794052278633537147003058712006013369023777892213640630941397833002710519641247944018371e-8
17
17
$x_k$:
4.871345193674403088349276556620113770860321241460771915159043858232258818564099273842429623481967559
17
17
$w_k$:
4.580578930798633305808892812223474448410255483561813165675248451766693930164575359806371141933901174e-11
18
1
$x_k$:
-5.048364008874466768372037578853652121096451147833483636368207771662991247777689837090730863055897968
18
1
$w_k$:
7.828199772115891029251474710119984043672573047094982333814686223502261125421041041634153690626306273e-12
18
2
$x_k$:
-4.248117873568126463023420160902081245461348125957106255695274514756009649652527950032642708427646145
18
2
$w_k$:
1.046720579579208244435596084350981982075576115628969893109044285549015429355427053726488731871180644e-8
18
3
$x_k$:
-3.573769068486266079500675993771889457939310976680155072886363189116486209988839048485470234740910876
18
3
$w_k$:
0.000001810654481093430409597023859106828585995362339688657369430991434852416986510401920908545444912638314
18
4
$x_k$:
-2.961377505531606844778632549061838214661398843053088824612273625677637206530905282001667011560885410
18
4
$w_k$:
0.00009181126867929403529146754073713004033606922667471140275107533618282143019839706289126834564390325538
18
5
$x_k$:
-2.386299089166686000264593014239945136707033332847681335596386472322516415507986322678421534625128034
18
5
$w_k$:
0.001888522630268417894381753254256556907551551741023170804421271500941788608683296067328677256124227148
18
6
$x_k$:
-1.835531604261628892253839444090601149266993636725739104328183139608537906437051657871282891068845728
18
6
$w_k$:
0.01864004238754465192193152219729959008832431611497441055249686788372157847670390802383385116497900487
18
7
$x_k$:
-1.300920858389617365666265554392610580218134639661226522772309775882782630084141194539623631652544514
18
7
$w_k$:
0.09730174764131542933085372341554007549247739677697774951826749430437693729326736208690119224933753943
18
8
$x_k$:
-0.7766829192674116613166594622838522947287132144905022620603101105022207649898232480775800188219080407
18
8
$w_k$:
0.2848072856699795785956068207126704288090226844471107751079824812113499343884272875275634496665001500
18
9
$x_k$:
-0.2582677505190967592581160987105796330017151353277434324130861043168057164979954832771497681678055643
18
9
$w_k$:
0.4834956947254555528764105221409973850125369095926593287268213602741901879675927748373444028347211230
18
10
$x_k$:
0.2582677505190967592581160987105796330017151353277434324130861043168057164979954832771497681678055643
18
10
$w_k$:
0.4834956947254555528764105221409973850125369095926593287268213602741901879675927748373444028347211230
18
11
$x_k$:
0.7766829192674116613166594622838522947287132144905022620603101105022207649898232480775800188219080407
18
11
$w_k$:
0.2848072856699795785956068207126704288090226844471107751079824812113499343884272875275634496665001500
18
12
$x_k$:
1.300920858389617365666265554392610580218134639661226522772309775882782630084141194539623631652544514
18
12
$w_k$:
0.09730174764131542933085372341554007549247739677697774951826749430437693729326736208690119224933753943
18
13
$x_k$:
1.835531604261628892253839444090601149266993636725739104328183139608537906437051657871282891068845728
18
13
$w_k$:
0.01864004238754465192193152219729959008832431611497441055249686788372157847670390802383385116497900487
18
14
$x_k$:
2.386299089166686000264593014239945136707033332847681335596386472322516415507986322678421534625128034
18
14
$w_k$:
0.001888522630268417894381753254256556907551551741023170804421271500941788608683296067328677256124227148
18
15
$x_k$:
2.961377505531606844778632549061838214661398843053088824612273625677637206530905282001667011560885410
18
15
$w_k$:
0.00009181126867929403529146754073713004033606922667471140275107533618282143019839706289126834564390325538
18
16
$x_k$:
3.573769068486266079500675993771889457939310976680155072886363189116486209988839048485470234740910876
18
16
$w_k$:
0.000001810654481093430409597023859106828585995362339688657369430991434852416986510401920908545444912638314
18
17
$x_k$:
4.248117873568126463023420160902081245461348125957106255695274514756009649652527950032642708427646145
18
17
$w_k$:
1.046720579579208244435596084350981982075576115628969893109044285549015429355427053726488731871180644e-8
18
18
$x_k$:
5.048364008874466768372037578853652121096451147833483636368207771662991247777689837090730863055897968
18
18
$w_k$:
7.828199772115891029251474710119984043672573047094982333814686223502261125421041041634153690626306273e-12
19
1
$x_k$:
-5.220271690537482164609671425002675737533884142119650485010159201139542982376197454928676210598512836
19
1
$w_k$:
1.326297094498515751852891543849668344374307664403707023261799240680543007575098742634125762632916158e-12
19
2
$x_k$:
-4.428532806603779437234985322258933539436258228041961771169150060215920944202267570774827634424630990
19
2
$w_k$:
2.163051009863554750196930772212759875203092045498277423467477646008840018440214967293409942893899679e-9
19
3
$x_k$:
-3.762187351964020097514893941035917958685813528333513546029782584435979944865509012998051086989419187
19
3
$w_k$:
4.488243147223122951794479155937697403119258765949707649554583048214324345192294109452114432051279698e-7
19
4
$x_k$:
-3.157848818347602281843180341199098258223742460115360590707322582128121788271797738853678675934376989
19
4
$w_k$:
0.00002720919776316162577119410252137364996233241621969425029795086551701374777697330644540855763966896560
19
5
$x_k$:
-2.591133789794542564921280841115634075192607383882508618459427360342975187324203602915772838181645658
19
5
$w_k$:
0.0006708775214071811061946962821000700491008944684511244892144367207163160585313769492689052587813522156
19
6
$x_k$:
-2.049231709850619375750508386686759734569276265553627389654501809105045977589177916962942035926641960
19
6
$w_k$:
0.007988866777722990209222114918611941650246961556111695260857374929910820234302320644173396747456164009
19
7
$x_k$:
-1.524170619393533031833548593665885298995383329479086227131013065455413707466149873057825626456712034
19
7
$w_k$:
0.05081038690905206735699081103581655698267021408018685336788331816710643371758090029140319001958496109
19
8
$x_k$:
-1.010368387134311351368598737261497478518548746022801059344748722710843614587764729585626426736759186
19
8
$w_k$:
0.1836327013069970741561484857657583633840087270003941532930412011855612151411949002264367143767969448
19
9
$x_k$:
-0.5035201634238882093738117650496884001578101358649439871913485578082990414255455922131780985270068161
19
9
$w_k$:
0.3916089886130302445040423136212175088340425300581154855465914053570994490929706046719405703119140187
19
10
$x_k$:
0
equals: Zero
comment: $x_{10}=0$, the central node of every rule of odd order
19
10
$w_k$:
0.5029748882761865308407313610956950386447864260167432195311442435030002552424078678412122636992238983
19
11
$x_k$:
0.5035201634238882093738117650496884001578101358649439871913485578082990414255455922131780985270068161
19
11
$w_k$:
0.3916089886130302445040423136212175088340425300581154855465914053570994490929706046719405703119140187
19
12
$x_k$:
1.010368387134311351368598737261497478518548746022801059344748722710843614587764729585626426736759186
19
12
$w_k$:
0.1836327013069970741561484857657583633840087270003941532930412011855612151411949002264367143767969448
19
13
$x_k$:
1.524170619393533031833548593665885298995383329479086227131013065455413707466149873057825626456712034
19
13
$w_k$:
0.05081038690905206735699081103581655698267021408018685336788331816710643371758090029140319001958496109
19
14
$x_k$:
2.049231709850619375750508386686759734569276265553627389654501809105045977589177916962942035926641960
19
14
$w_k$:
0.007988866777722990209222114918611941650246961556111695260857374929910820234302320644173396747456164009
19
15
$x_k$:
2.591133789794542564921280841115634075192607383882508618459427360342975187324203602915772838181645658
19
15
$w_k$:
0.0006708775214071811061946962821000700491008944684511244892144367207163160585313769492689052587813522156
19
16
$x_k$:
3.157848818347602281843180341199098258223742460115360590707322582128121788271797738853678675934376989
19
16
$w_k$:
0.00002720919776316162577119410252137364996233241621969425029795086551701374777697330644540855763966896560
19
17
$x_k$:
3.762187351964020097514893941035917958685813528333513546029782584435979944865509012998051086989419187
19
17
$w_k$:
4.488243147223122951794479155937697403119258765949707649554583048214324345192294109452114432051279698e-7
19
18
$x_k$:
4.428532806603779437234985322258933539436258228041961771169150060215920944202267570774827634424630990
19
18
$w_k$:
2.163051009863554750196930772212759875203092045498277423467477646008840018440214967293409942893899679e-9
19
19
$x_k$:
5.220271690537482164609671425002675737533884142119650485010159201139542982376197454928676210598512836
19
19
$w_k$:
1.326297094498515751852891543849668344374307664403707023261799240680543007575098742634125762632916158e-12
20
1
$x_k$:
-5.387480890011232862016900410681120753996286449065914889735765329809765144039123345546418701033761009
20
1
$w_k$:
2.229393645534151292522500616029095784862440697814397468208086324913139914789474387783763159110631306e-13
20
2
$x_k$:
-4.603682449550744273077675248978347585113398487761923653562330942797725663212287593074492568163350331
20
2
$w_k$:
4.399340992273180553628851455467928215614956748546294327205244869356949006244434454298947376041041204e-10
20
3
$x_k$:
-3.944764040115625210375628800524411807149768127888313020647662390117733086334323933618415495086608911
20
3
$w_k$:
1.086069370769281693999524563447163431184931585602462221833075806600293260837414708790214759229478695e-7
20
4
$x_k$:
-3.347854567383216326914924522996463698510478590293677056533989972309375156763466662250493409956746235
20
4
$w_k$:
0.000007802556478532063694145991999647569101115074627663388830002921718775115240632958232520074968557174312
20
5
$x_k$:
-2.788806058428130480525033756403185410670698887902043973964712799167655237419371539612921795639174059
20
5
$w_k$:
0.0002283386360163539672571459179634955392289055793054606789823413087214651932021947862046454263347545769
20
6
$x_k$:
-2.254974002089275523082333344734565128082265316026396151312422194351620392351783506947663365752493264
20
6
$w_k$:
0.003243773342237861832183247132353705442756465763937999983145703801372592897587625735589935505265718170
20
7
$x_k$:
-1.738537712116586206780865662136406442951409419600399894464818572382479821379739558835074388182530879
20
7
$w_k$:
0.02481052088746361088216495255894039440066580288108799116611394493128600246865465969360723118742492058
20
8
$x_k$:
-1.234076215395323007885818346959410229585445930069362387126965421690370282312391223717487006085546077
20
8
$w_k$:
0.1090172060200233200137550335354255770842587925928164541648600405807818935465051822643823445958903827
20
9
$x_k$:
-0.7374737285453943587056051442521042290772162039767941872968422099956257623371318234537355646571045775
20
9
$w_k$:
0.2866755053628341297196597062280879168237637379633907410050778172615389902289220923305626988960803559
20
10
$x_k$:
-0.2453407083009012499038365306336166239661338513034857348785924037753487947015036755489007014871859258
20
10
$w_k$:
0.4622436696006100896503286398612081142142610585728867745492609883461099672478914045615026807931586068
20
11
$x_k$:
0.2453407083009012499038365306336166239661338513034857348785924037753487947015036755489007014871859258
20
11
$w_k$:
0.4622436696006100896503286398612081142142610585728867745492609883461099672478914045615026807931586068
20
12
$x_k$:
0.7374737285453943587056051442521042290772162039767941872968422099956257623371318234537355646571045775
20
12
$w_k$:
0.2866755053628341297196597062280879168237637379633907410050778172615389902289220923305626988960803559
20
13
$x_k$:
1.234076215395323007885818346959410229585445930069362387126965421690370282312391223717487006085546077
20
13
$w_k$:
0.1090172060200233200137550335354255770842587925928164541648600405807818935465051822643823445958903827
20
14
$x_k$:
1.738537712116586206780865662136406442951409419600399894464818572382479821379739558835074388182530879
20
14
$w_k$:
0.02481052088746361088216495255894039440066580288108799116611394493128600246865465969360723118742492058
20
15
$x_k$:
2.254974002089275523082333344734565128082265316026396151312422194351620392351783506947663365752493264
20
15
$w_k$:
0.003243773342237861832183247132353705442756465763937999983145703801372592897587625735589935505265718170
20
16
$x_k$:
2.788806058428130480525033756403185410670698887902043973964712799167655237419371539612921795639174059
20
16
$w_k$:
0.0002283386360163539672571459179634955392289055793054606789823413087214651932021947862046454263347545769
20
17
$x_k$:
3.347854567383216326914924522996463698510478590293677056533989972309375156763466662250493409956746235
20
17
$w_k$:
0.000007802556478532063694145991999647569101115074627663388830002921718775115240632958232520074968557174312
20
18
$x_k$:
3.944764040115625210375628800524411807149768127888313020647662390117733086334323933618415495086608911
20
18
$w_k$:
1.086069370769281693999524563447163431184931585602462221833075806600293260837414708790214759229478695e-7
20
19
$x_k$:
4.603682449550744273077675248978347585113398487761923653562330942797725663212287593074492568163350331
20
19
$w_k$:
4.399340992273180553628851455467928215614956748546294327205244869356949006244434454298947376041041204e-10
20
20
$x_k$:
5.387480890011232862016900410681120753996286449065914889735765329809765144039123345546418701033761009
20
20
$w_k$:
2.229393645534151292522500616029095784862440697814397468208086324913139914789474387783763159110631306e-13
21
1
$x_k$:
-5.550351873264678245229686877101383013327951067300069097918005588393649556960247053832200227886686035
21
1
$w_k$:
3.720365070136049262158575012570201035460298400582835548467526944205156148105405578696743352607290899e-14
21
2
$x_k$:
-4.773992343411219429701509577115341560120602906396796338938985428969153907042225223482889102246865146
21
2
$w_k$:
8.818611242049951594159495320098400829076384529934793384366952258967762643728745003866586561690217987e-11
21
3
$x_k$:
-4.121995547491840020816900677276602323107881634257438625589153425029489013631454193345605578920449555
21
3
$w_k$:
2.571230180059313704775587623445262143978748229996475858247920078396422082935143255323862110235532942e-8
21
4
$x_k$:
-3.531972877137677739171383282622677740692432878337947900292316187858927980080903158524730370961760760
21
4
$w_k$:
0.000002171884898056669582873498368692744695050814775444331406967575398267025461327737111029422751907682960
21
5
$x_k$:
-2.979991207704598002537727817527695160851492220056180439793240747175811411193645116692437766422962033
21
5
$w_k$:
0.00007478398867310061169097859951377277019473114077962401155510816835147573743607924835089741239163821503
21
6
$x_k$:
-2.453552124512838002000735406157451726337018643940252793596985887732825445776109647594812609442147722
21
6
$w_k$:
0.001254982041726410545852102357257876980882152781770834617295123351554977046148823616777217512227500490
21
7
$x_k$:
-1.944962949186253841901916715473994175832571363289334270345692776035907400632706160248737291373533506
21
7
$w_k$:
0.01141406583743438337658450472868748996844156589792369105640983007702474903457351402326295595932020767
21
8
$x_k$:
-1.448934250650731962657293148676816502651559429177050744380997878912278644163323290209125345845693973
21
8
$w_k$:
0.06017964665891226717166417928118484245383506842925057204576183290017164040593135051092649686083420118
21
9
$x_k$:
-0.9614996344183690642794222713524960841591128536615060508621122056726343993785626626224242387847603457
21
9
$w_k$:
0.1921203240669977561290824607392939519616049314898744548977968766949495697818441331970541198673332355
21
10
$x_k$:
-0.4794507070791075762945981035134211586268134086066472405547413711828880620859613844492130400540735501
21
10
$w_k$:
0.3816690736135020982704166415639374584273965412197690427408780764792212884682051798413095236683759455
21
11
$x_k$:
0
equals: Zero
comment: $x_{11}=0$, the central node of every rule of odd order
21
11
$w_k$:
0.4790237031201776484197441534244714653759870723968983043153754700028573859451503503249640606659275222
21
12
$x_k$:
0.4794507070791075762945981035134211586268134086066472405547413711828880620859613844492130400540735501
21
12
$w_k$:
0.3816690736135020982704166415639374584273965412197690427408780764792212884682051798413095236683759455
21
13
$x_k$:
0.9614996344183690642794222713524960841591128536615060508621122056726343993785626626224242387847603457
21
13
$w_k$:
0.1921203240669977561290824607392939519616049314898744548977968766949495697818441331970541198673332355
21
14
$x_k$:
1.448934250650731962657293148676816502651559429177050744380997878912278644163323290209125345845693973
21
14
$w_k$:
0.06017964665891226717166417928118484245383506842925057204576183290017164040593135051092649686083420118
21
15
$x_k$:
1.944962949186253841901916715473994175832571363289334270345692776035907400632706160248737291373533506
21
15
$w_k$:
0.01141406583743438337658450472868748996844156589792369105640983007702474903457351402326295595932020767
21
16
$x_k$:
2.453552124512838002000735406157451726337018643940252793596985887732825445776109647594812609442147722
21
16
$w_k$:
0.001254982041726410545852102357257876980882152781770834617295123351554977046148823616777217512227500490
21
17
$x_k$:
2.979991207704598002537727817527695160851492220056180439793240747175811411193645116692437766422962033
21
17
$w_k$:
0.00007478398867310061169097859951377277019473114077962401155510816835147573743607924835089741239163821503
21
18
$x_k$:
3.531972877137677739171383282622677740692432878337947900292316187858927980080903158524730370961760760
21
18
$w_k$:
0.000002171884898056669582873498368692744695050814775444331406967575398267025461327737111029422751907682960
21
19
$x_k$:
4.121995547491840020816900677276602323107881634257438625589153425029489013631454193345605578920449555
21
19
$w_k$:
2.571230180059313704775587623445262143978748229996475858247920078396422082935143255323862110235532942e-8
21
20
$x_k$:
4.773992343411219429701509577115341560120602906396796338938985428969153907042225223482889102246865146
21
20
$w_k$:
8.818611242049951594159495320098400829076384529934793384366952258967762643728745003866586561690217987e-11
21
21
$x_k$:
5.550351873264678245229686877101383013327951067300069097918005588393649556960247053832200227886686035
21
21
$w_k$:
3.720365070136049262158575012570201035460298400582835548467526944205156148105405578696743352607290899e-14
22
1
$x_k$:
-5.709201353205263777359343078116754163221696227119705021387766625550905871861418869453449214110470023
22
1
$w_k$:
6.167183424404048835277030337205963930442225055022651971237772103584804180068801148154631965007127251e-15
22
2
$x_k$:
-4.939834131060175887870213251420454116377454482668662881299621435773415000054254649725124616454437730
22
2
$w_k$:
1.744339007547992844283184605288232059823383933966054043323721785642492757544537234836913856452989914e-11
22
3
$x_k$:
-4.294312480593161515750307940598903517065375012394862184274500507791315406845231736816374615483230581
22
3
$w_k$:
5.966990986059652717281370415326945453924456678726127331091931500413921478037633481299208215186207282e-9
22
4
$x_k$:
-3.710701532877804914809177572909852375668432003169331865080623723962894080890846540806174824915650088
22
4
$w_k$:
5.884287563301005780415851103731181156540520976538707094115860742556764153581739884219552552346537338e-7
22
5
$x_k$:
-3.165265909202137446987186492058512870355694711612836761620414434709049372970890661649603683852348574
22
5
$w_k$:
0.00002365512855251045804735374404097224229622940465205482416308631687100848099189880597383453831676289308
22
6
$x_k$:
-2.645637441058172700798427239008334936690575108382177466423468501301629244208739142742374169842347398
22
6
$w_k$:
0.0004648850508842522441820254931828412007850601276619914592523681946581478238304236303326369910940046939
22
7
$x_k$:
-2.144233592798534463619642606542287013057269311460678039264537262972835637464917657510121273759736853
22
7
$w_k$:
0.004978399335051647404741388773770051256697952405129905157264963668836014049642680535009352471659059564
22
8
$x_k$:
-1.655874373286422495437710109710877859787444691773889959071583053244005400121839769938984400204752044
22
8
$w_k$:
0.03114037088442384830069932154877271569762054582223095921762380394149254434950711351661882488974774944
22
9
$x_k$:
-1.176713958481244447422825266669440157037205599825477722260337409880404064538405712836934685509619976
22
9
$w_k$:
0.1191023609587824659742524433452390610336038886137826984271018942499552439084660484154109091393408378
22
10
$x_k$:
-0.7036860971700069321664484320460196480793213673378048344184437643180608793098724551386471506214215227
22
10
$w_k$:
0.2869714332469071144010214792214957364185402793186677710117172511408672583986184885456321150261902760
22
11
$x_k$:
-0.2341791399309906350983107826836416058305566122231184123617864002685317702872977757153246133996255286
22
11
$w_k$:
0.4435452264349593014470046381859233020209501378303495698482282216505612135655940886472503109089428993
22
12
$x_k$:
0.2341791399309906350983107826836416058305566122231184123617864002685317702872977757153246133996255286
22
12
$w_k$:
0.4435452264349593014470046381859233020209501378303495698482282216505612135655940886472503109089428993
22
13
$x_k$:
0.7036860971700069321664484320460196480793213673378048344184437643180608793098724551386471506214215227
22
13
$w_k$:
0.2869714332469071144010214792214957364185402793186677710117172511408672583986184885456321150261902760
22
14
$x_k$:
1.176713958481244447422825266669440157037205599825477722260337409880404064538405712836934685509619976
22
14
$w_k$:
0.1191023609587824659742524433452390610336038886137826984271018942499552439084660484154109091393408378
22
15
$x_k$:
1.655874373286422495437710109710877859787444691773889959071583053244005400121839769938984400204752044
22
15
$w_k$:
0.03114037088442384830069932154877271569762054582223095921762380394149254434950711351661882488974774944
22
16
$x_k$:
2.144233592798534463619642606542287013057269311460678039264537262972835637464917657510121273759736853
22
16
$w_k$:
0.004978399335051647404741388773770051256697952405129905157264963668836014049642680535009352471659059564
22
17
$x_k$:
2.645637441058172700798427239008334936690575108382177466423468501301629244208739142742374169842347398
22
17
$w_k$:
0.0004648850508842522441820254931828412007850601276619914592523681946581478238304236303326369910940046939
22
18
$x_k$:
3.165265909202137446987186492058512870355694711612836761620414434709049372970890661649603683852348574
22
18
$w_k$:
0.00002365512855251045804735374404097224229622940465205482416308631687100848099189880597383453831676289308
22
19
$x_k$:
3.710701532877804914809177572909852375668432003169331865080623723962894080890846540806174824915650088
22
19
$w_k$:
5.884287563301005780415851103731181156540520976538707094115860742556764153581739884219552552346537338e-7
22
20
$x_k$:
4.294312480593161515750307940598903517065375012394862184274500507791315406845231736816374615483230581
22
20
$w_k$:
5.966990986059652717281370415326945453924456678726127331091931500413921478037633481299208215186207282e-9
22
21
$x_k$:
4.939834131060175887870213251420454116377454482668662881299621435773415000054254649725124616454437730
22
21
$w_k$:
1.744339007547992844283184605288232059823383933966054043323721785642492757544537234836913856452989914e-11
22
22
$x_k$:
5.709201353205263777359343078116754163221696227119705021387766625550905871861418869453449214110470023
22
22
$w_k$:
6.167183424404048835277030337205963930442225055022651971237772103584804180068801148154631965007127251e-15
23
1
$x_k$:
-5.864309498984572565387484134743478075275255392598424293967043510584874620519272560464673755568192374
23
1
$w_k$:
1.016038462063672304561172900551674074832577025739703699389965841889313840264428769964810561960954261e-15
23
2
$x_k$:
-5.101534610476677129687497661653837570891772033007854788871993067097958633012401734454883439006847780
23
2
$w_k$:
3.408314098030539016678711120531771766052629723008548198822653009057644552883337246383535983047914722e-12
23
3
$x_k$:
-4.462091173740006676731861570711616670507188051104049701226599062497235604224855854749770839802060118
23
3
$w_k$:
1.359629650402887767082084199376572505895972740953685002801654113577815951862809524612311314801335362e-9
23
4
$x_k$:
-3.884472708106101866072487602876231131846071090382229627823180787092550538553375210277139847481713300
23
4
$w_k$:
1.555339329145766322011779994622807527945675349407353176020564536289188742650960664605254557587377913e-7
23
5
$x_k$:
-3.345127159941224572474398145852147638214501244486729269321107204920691276543079109100389447717306792
23
5
$w_k$:
0.000007249295918002257039559167883091269249357935513353812937875313520570479433938635662188878082815197079
23
6
$x_k$:
-2.831803787126156901448061407339466927739469591246370375005536301143768536050754383013744559431176883
23
6
$w_k$:
0.0001655616991418743955745710622118755878919919503593020634492502092374963732019519498101005194799838746
23
7
$x_k$:
-2.337016211474455786446235021735346304109093620751439965873762629935691421032394582095135258946979830
23
7
$w_k$:
0.002069567874960638158073836662423533859009494967037799713751210481089218799534808198047806837592785057
23
8
$x_k$:
-1.855677037671371062515047537180864310362575368408908756061804164132459415339876911440413450531442035
23
8
$w_k$:
0.01520708400448412962451276792623873362270314363132372467627051179696930565565877238243656474182943992
23
9
$x_k$:
-1.384039585682495237326347171180129719039064510229247435096070897935182250094400390664482621926728384
23
9
$w_k$:
0.06889028942908731698824068316409549463890956561629693757195452333804372627703874920321671756456107881
23
10
$x_k$:
-0.9191514654425637654317192395927129320836511496417918849121840521775044408434726387269813109585069411
23
10
$w_k$:
0.1986448985780225957938544822657124321826015233741246550099347244500598041242507533524759000867091733
23
11
$x_k$:
-0.4585383500681047977578873292836384809634038522357625396390926956246603613020453872311464509146200689
23
11
$w_k$:
0.3721438248775648598976367623134106650208756007539512843361284644016261157497354594630697382617833194
23
12
$x_k$:
0
equals: Zero
comment: $x_{12}=0$, the central node of every rule of odd order
23
12
$w_k$:
0.4581965855932134028362770163190596625335528518579027258668808843505592387301438133543134493326263256
23
13
$x_k$:
0.4585383500681047977578873292836384809634038522357625396390926956246603613020453872311464509146200689
23
13
$w_k$:
0.3721438248775648598976367623134106650208756007539512843361284644016261157497354594630697382617833194
23
14
$x_k$:
0.9191514654425637654317192395927129320836511496417918849121840521775044408434726387269813109585069411
23
14
$w_k$:
0.1986448985780225957938544822657124321826015233741246550099347244500598041242507533524759000867091733
23
15
$x_k$:
1.384039585682495237326347171180129719039064510229247435096070897935182250094400390664482621926728384
23
15
$w_k$:
0.06889028942908731698824068316409549463890956561629693757195452333804372627703874920321671756456107881
23
16
$x_k$:
1.855677037671371062515047537180864310362575368408908756061804164132459415339876911440413450531442035
23
16
$w_k$:
0.01520708400448412962451276792623873362270314363132372467627051179696930565565877238243656474182943992
23
17
$x_k$:
2.337016211474455786446235021735346304109093620751439965873762629935691421032394582095135258946979830
23
17
$w_k$:
0.002069567874960638158073836662423533859009494967037799713751210481089218799534808198047806837592785057
23
18
$x_k$:
2.831803787126156901448061407339466927739469591246370375005536301143768536050754383013744559431176883
23
18
$w_k$:
0.0001655616991418743955745710622118755878919919503593020634492502092374963732019519498101005194799838746
23
19
$x_k$:
3.345127159941224572474398145852147638214501244486729269321107204920691276543079109100389447717306792
23
19
$w_k$:
0.000007249295918002257039559167883091269249357935513353812937875313520570479433938635662188878082815197079
23
20
$x_k$:
3.884472708106101866072487602876231131846071090382229627823180787092550538553375210277139847481713300
23
20
$w_k$:
1.555339329145766322011779994622807527945675349407353176020564536289188742650960664605254557587377913e-7
23
21
$x_k$:
4.462091173740006676731861570711616670507188051104049701226599062497235604224855854749770839802060118
23
21
$w_k$:
1.359629650402887767082084199376572505895972740953685002801654113577815951862809524612311314801335362e-9
23
22
$x_k$:
5.101534610476677129687497661653837570891772033007854788871993067097958633012401734454883439006847780
23
22
$w_k$:
3.408314098030539016678711120531771766052629723008548198822653009057644552883337246383535983047914722e-12
23
23
$x_k$:
5.864309498984572565387484134743478075275255392598424293967043510584874620519272560464673755568192374
23
23
$w_k$:
1.016038462063672304561172900551674074832577025739703699389965841889313840264428769964810561960954261e-15
24
1
$x_k$:
-6.015925561425739717348573508988387865104133615477457492639963635179314594443733990597688362325365488
24
1
$w_k$:
1.664368496489108873772625344608517684889896475468920172331477487701247371411081104099994588354487574e-16
24
2
$x_k$:
-5.259382927668044367430723043983956414150083834008439074164538661442218458358068552641473904301135445
24
2
$w_k$:
6.584620243078170064562059086818197338523290303110283591989197683575370303254595241503623135924829351e-13
24
3
$x_k$:
-4.625662756423787265048649237762599899693337585502157825453086781242231858123617158801294149967047593
24
3
$w_k$:
3.046254269987563903885915972870225943343516307630359680241994365676533187143878491335544116390695156e-10
24
4
$x_k$:
-4.053664402448149503947662979232829670051415483848170761835530445662933217016873891301004222067988870
24
4
$w_k$:
4.018971174941429684540212806977075774639196777143103783683310775403509887179979743651753158756366241e-8
24
5
$x_k$:
-3.520006813034524711289872276085998470763578155004754058640199340166825021556774970637665121759008328
24
5
$w_k$:
0.000002158245704902333632238640097038978868163400604063995816382013494381918345075402453727854860919740136
24
6
$x_k$:
-3.012546137565564825654538584205116643304927506452818303640960629592366530134908147529687325060705447
24
6
$w_k$:
0.00005688691636404379769041846360589869542853819488804586999249267724179923730668613369362676253666590761
24
7
$x_k$:
-2.523881017011426974199076023325426339709327238944469227568834152130637376356499266009656235891321019
24
7
$w_k$:
0.0008236924826884174579183242203078475385205936047898107214197131750406334882587534173604521094056289647
24
8
$x_k$:
-2.049003573661698911787083995324292539869849124528368807719070360238896910818776503099074917478843477
24
8
$w_k$:
0.007048355810072670970003755975798856042855264255688133950679297742044454406601100389234457080215252249
24
9
$x_k$:
-1.584250010961694148505633362017265479983735228703683601199990338069120780485881669026943618932069452
24
9
$w_k$:
0.03744547050323074601333012963655052102254271156629548162646905472894714834937558750219184149091541671
24
10
$x_k$:
-1.126760817611245072133061267730146446386023934345916594084784609402717171762870079552521433861140844
24
10
$w_k$:
0.1277396217845591606472550262433483059082646489412772335839446181713973593576903244965970594998059184
24
11
$x_k$:
-0.6741711070372122360002459237300052475272839880367550185996087636761075573086114644760069886711188838
24
11
$w_k$:
0.2861795353464430179018952387471771370471300849624201919090460646682860664085774381784849943724585794
24
12
$x_k$:
-0.2244145474725155851511367155265091209943196130741495354506907140242976180299415793820504397658641858
24
12
$w_k$:
0.4269311638686992496531479080356237359361681018232903822355038916414804556105026274982112829157271799
24
13
$x_k$:
0.2244145474725155851511367155265091209943196130741495354506907140242976180299415793820504397658641858
24
13
$w_k$:
0.4269311638686992496531479080356237359361681018232903822355038916414804556105026274982112829157271799
24
14
$x_k$:
0.6741711070372122360002459237300052475272839880367550185996087636761075573086114644760069886711188838
24
14
$w_k$:
0.2861795353464430179018952387471771370471300849624201919090460646682860664085774381784849943724585794
24
15
$x_k$:
1.126760817611245072133061267730146446386023934345916594084784609402717171762870079552521433861140844
24
15
$w_k$:
0.1277396217845591606472550262433483059082646489412772335839446181713973593576903244965970594998059184
24
16
$x_k$:
1.584250010961694148505633362017265479983735228703683601199990338069120780485881669026943618932069452
24
16
$w_k$:
0.03744547050323074601333012963655052102254271156629548162646905472894714834937558750219184149091541671
24
17
$x_k$:
2.049003573661698911787083995324292539869849124528368807719070360238896910818776503099074917478843477
24
17
$w_k$:
0.007048355810072670970003755975798856042855264255688133950679297742044454406601100389234457080215252249
24
18
$x_k$:
2.523881017011426974199076023325426339709327238944469227568834152130637376356499266009656235891321019
24
18
$w_k$:
0.0008236924826884174579183242203078475385205936047898107214197131750406334882587534173604521094056289647
24
19
$x_k$:
3.012546137565564825654538584205116643304927506452818303640960629592366530134908147529687325060705447
24
19
$w_k$:
0.00005688691636404379769041846360589869542853819488804586999249267724179923730668613369362676253666590761
24
20
$x_k$:
3.520006813034524711289872276085998470763578155004754058640199340166825021556774970637665121759008328
24
20
$w_k$:
0.000002158245704902333632238640097038978868163400604063995816382013494381918345075402453727854860919740136
24
21
$x_k$:
4.053664402448149503947662979232829670051415483848170761835530445662933217016873891301004222067988870
24
21
$w_k$:
4.018971174941429684540212806977075774639196777143103783683310775403509887179979743651753158756366241e-8
24
22
$x_k$:
4.625662756423787265048649237762599899693337585502157825453086781242231858123617158801294149967047593
24
22
$w_k$:
3.046254269987563903885915972870225943343516307630359680241994365676533187143878491335544116390695156e-10
24
23
$x_k$:
5.259382927668044367430723043983956414150083834008439074164538661442218458358068552641473904301135445
24
23
$w_k$:
6.584620243078170064562059086818197338523290303110283591989197683575370303254595241503623135924829351e-13
24
24
$x_k$:
6.015925561425739717348573508988387865104133615477457492639963635179314594443733990597688362325365488
24
24
$w_k$:
1.664368496489108873772625344608517684889896475468920172331477487701247371411081104099994588354487574e-16
25
1
$x_k$:
-6.164272434052451770932480843404636405626445377584670104106973420941438388290671767973207929826854962
25
1
$w_k$:
2.711923514038411885272141780282895043639796690294998499623209415251932625069446579917117006197675726e-17
25
2
$x_k$:
-5.413636355280033809666868049956050517582030947755807270861488662355108046574327556813616318981655625
25
2
$w_k$:
1.258814987746546119186693449705802650740399855259852594425063701382505501189740805013670610068604194e-13
25
3
$x_k$:
-4.785320367352224057257145668598121119303088428257808031198717376527656286644445983434730790291654583
25
3
$w_k$:
6.719638417706208255859183808133762086663379877375453835769498150017841886789142922357455073857394258e-11
25
4
$x_k$:
-4.218609444386561381439023915842375668642672535114194252487026873375934087708716212434775323541348620
25
4
$w_k$:
1.017038250301846044019599940034423491801473950998456713811261898603570605557359695772639317590762991e-8
25
5
$x_k$:
-3.690282876998355901627081542822832157074389147731656443112496131475414969690190816599905154704150574
25
5
$w_k$:
6.257032499691111620479845087281227475451211804850396892301671204661625327722268899084618494119576278e-7
25
6
$x_k$:
-3.188294924425104712159545009770331076899937960831357971340773009831682198505549728108406014605743615
25
6
$w_k$:
0.00001891597295734051231572574215261311791540546039853893027016934911302169847127906784217232788590383066
25
7
$x_k$:
-2.705320237173025993251557164122168553383205254315171969642826068209593596109449003403112176423161234
25
7
$w_k$:
0.0003150836387454841148003386632446259495246970200445258559581411889490361314407487423631973255620899951
25
8
$x_k$:
-2.236420130267280870203105756980660874995024838565774922516132722923556247646539888232576135566698855
25
8
$w_k$:
0.003115708720125633461971533313284228904880019436278084046316612958134547427274129305523225585890813490
25
9
$x_k$:
-1.778001124337147458196308669921803544105784793905261719963935320367246962790878615417999157137980426
25
9
$w_k$:
0.01924309896540889549702233483619141920146715146134724116236396314525448778890727309730042921094828299
25
10
$x_k$:
-1.327280702073083951787488518410767251295691968011094485814544718590436091367673501080102622675414936
25
10
$w_k$:
0.07688899517580878575354762017263019145826886939232249352025257468343791075904610524879129782188959534
25
11
$x_k$:
-0.8819827562138213563716092246559304920009681294602677544487258422655788603373973148083001041687837497
25
11
$w_k$:
0.2036211366781240505655204558383041917867862564725238507242132148224704559200446096139303167457895448
25
12
$x_k$:
-0.4401472986453083045786123915068033177513967559783734481883583061531099559485641506462983917354109176
25
12
$w_k$:
0.3630889892758906254577983995364791943491184928299072512699225994439658116792132654638307134848611612
25
13
$x_k$:
0
equals: Zero
comment: $x_{13}=0$, the central node of every rule of odd order
25
13
$w_k$:
0.4398687221694848667228259356662972760322107377835866168322056489765368691809380608201409113593212725
25
14
$x_k$:
0.4401472986453083045786123915068033177513967559783734481883583061531099559485641506462983917354109176
25
14
$w_k$:
0.3630889892758906254577983995364791943491184928299072512699225994439658116792132654638307134848611612
25
15
$x_k$:
0.8819827562138213563716092246559304920009681294602677544487258422655788603373973148083001041687837497
25
15
$w_k$:
0.2036211366781240505655204558383041917867862564725238507242132148224704559200446096139303167457895448
25
16
$x_k$:
1.327280702073083951787488518410767251295691968011094485814544718590436091367673501080102622675414936
25
16
$w_k$:
0.07688899517580878575354762017263019145826886939232249352025257468343791075904610524879129782188959534
25
17
$x_k$:
1.778001124337147458196308669921803544105784793905261719963935320367246962790878615417999157137980426
25
17
$w_k$:
0.01924309896540889549702233483619141920146715146134724116236396314525448778890727309730042921094828299
25
18
$x_k$:
2.236420130267280870203105756980660874995024838565774922516132722923556247646539888232576135566698855
25
18
$w_k$:
0.003115708720125633461971533313284228904880019436278084046316612958134547427274129305523225585890813490
25
19
$x_k$:
2.705320237173025993251557164122168553383205254315171969642826068209593596109449003403112176423161234
25
19
$w_k$:
0.0003150836387454841148003386632446259495246970200445258559581411889490361314407487423631973255620899951
25
20
$x_k$:
3.188294924425104712159545009770331076899937960831357971340773009831682198505549728108406014605743615
25
20
$w_k$:
0.00001891597295734051231572574215261311791540546039853893027016934911302169847127906784217232788590383066
25
21
$x_k$:
3.690282876998355901627081542822832157074389147731656443112496131475414969690190816599905154704150574
25
21
$w_k$:
6.257032499691111620479845087281227475451211804850396892301671204661625327722268899084618494119576278e-7
25
22
$x_k$:
4.218609444386561381439023915842375668642672535114194252487026873375934087708716212434775323541348620
25
22
$w_k$:
1.017038250301846044019599940034423491801473950998456713811261898603570605557359695772639317590762991e-8
25
23
$x_k$:
4.785320367352224057257145668598121119303088428257808031198717376527656286644445983434730790291654583
25
23
$w_k$:
6.719638417706208255859183808133762086663379877375453835769498150017841886789142922357455073857394258e-11
25
24
$x_k$:
5.413636355280033809666868049956050517582030947755807270861488662355108046574327556813616318981655625
25
24
$w_k$:
1.258814987746546119186693449705802650740399855259852594425063701382505501189740805013670610068604194e-13
25
25
$x_k$:
6.164272434052451770932480843404636405626445377584670104106973420941438388290671767973207929826854962
25
25
$w_k$:
2.711923514038411885272141780282895043639796690294998499623209415251932625069446579917117006197675726e-17
26
1
$x_k$:
-6.309550385625693500904546656983340584766600795725299795284066690924610828355619393478832132373439161
26
1
$w_k$:
4.396916094753901864568217063627761319905514268245765309184922267549846248109151510157237666697736972e-18
26
2
$x_k$:
-5.564524981950103097163064482378535941615149081748118204112946148628905909232343383432040025192071984
26
2
$w_k$:
2.383148659372160648946267218611284245698599016286779271153795050378592182605060113413992463313679339e-14
26
3
$x_k$:
-4.941324957241379073791445944818101470023455076116482353598476391395138404222856114191833459333242791
26
3
$w_k$:
1.460999933981612369422693926806920339077808228657174780423914522514881529343845557636000587655407090e-11
26
4
$x_k$:
-4.379602662983304971912542923699548675773315517468232635894148800810072768969364326989929465157323562
26
4
$w_k$:
2.524494034490533687075781476716255502286458470231026099265696894794611332260121675548250040482854483e-9
26
5
$x_k$:
-3.856288419909149021807326666736635804532328355906924610206256225623256327737141846214373098833200106
26
5
$w_k$:
1.770106337397356352685723437869194702907400038089468175830646464558217617515929752102371808072129998e-7
26
6
$x_k$:
-3.359427182350830136645671275339555710139120247832657881911014091405321426871475611221545137934418093
26
6
$w_k$:
0.000006103291717396013040283843449380876456778992506090816682044489459781447422376153319015876945395798395
26
7
$x_k$:
-2.881762219543087217483493311969365736033663238307335184492222063678660082518567758674173033815367676
26
7
$w_k$:
0.0001162297016031096300265745785068695023825309172462852534877102045895795451022158357847286014978636796
26
8
$x_k$:
-2.418415764773779147799016688510134219858822943813731678985794077412195211151124039399348650577625709
26
8
$w_k$:
0.001319064722323857351848329919579345012191233782062838567313007099651266381752547346246001645353849748
26
9
$x_k$:
-1.965854785641136575228750814656935800905675538277820585262087190056193466633881949098052212353268848
26
9
$w_k$:
0.009397901291159579588159472076172696180532431618985674547557861280004859583180371670028914640595709244
26
10
$x_k$:
-1.521361516651921366741423962559162513722808482227127587303669708453538743901682456758403367993972729
26
10
$w_k$:
0.04359822721725079772942178129440263461354845873055902848445103777808654656828364446907023881667202151
26
11
$x_k$:
-1.082733011077883356232554051114734186214721286745726461715202467584730958371300667097912544756752728
26
11
$w_k$:
0.1351133279117878517573042696914529257925354035871335494108721845098598786228724490710296903180138368
26
12
$x_k$:
-0.6480952139934483096825214382869983345884044683849276098246703650833481316044434154506860152209220149
26
12
$w_k$:
0.2846322411767845171608861886590269059627552032900782488726402157583388025307811744001285098959469116
26
13
$x_k$:
-0.2157778562434634346451580121953813729262493103199737178484004308526457087694148787361032476386184252
26
13
$w_k$:
0.4120436505903692949689019459052383033141745142551631807938072676334055846391733591236605885021702407
26
14
$x_k$:
0.2157778562434634346451580121953813729262493103199737178484004308526457087694148787361032476386184252
26
14
$w_k$:
0.4120436505903692949689019459052383033141745142551631807938072676334055846391733591236605885021702407
26
15
$x_k$:
0.6480952139934483096825214382869983345884044683849276098246703650833481316044434154506860152209220149
26
15
$w_k$:
0.2846322411767845171608861886590269059627552032900782488726402157583388025307811744001285098959469116
26
16
$x_k$:
1.082733011077883356232554051114734186214721286745726461715202467584730958371300667097912544756752728
26
16
$w_k$:
0.1351133279117878517573042696914529257925354035871335494108721845098598786228724490710296903180138368
26
17
$x_k$:
1.521361516651921366741423962559162513722808482227127587303669708453538743901682456758403367993972729
26
17
$w_k$:
0.04359822721725079772942178129440263461354845873055902848445103777808654656828364446907023881667202151
26
18
$x_k$:
1.965854785641136575228750814656935800905675538277820585262087190056193466633881949098052212353268848
26
18
$w_k$:
0.009397901291159579588159472076172696180532431618985674547557861280004859583180371670028914640595709244
26
19
$x_k$:
2.418415764773779147799016688510134219858822943813731678985794077412195211151124039399348650577625709
26
19
$w_k$:
0.001319064722323857351848329919579345012191233782062838567313007099651266381752547346246001645353849748
26
20
$x_k$:
2.881762219543087217483493311969365736033663238307335184492222063678660082518567758674173033815367676
26
20
$w_k$:
0.0001162297016031096300265745785068695023825309172462852534877102045895795451022158357847286014978636796
26
21
$x_k$:
3.359427182350830136645671275339555710139120247832657881911014091405321426871475611221545137934418093
26
21
$w_k$:
0.000006103291717396013040283843449380876456778992506090816682044489459781447422376153319015876945395798395
26
22
$x_k$:
3.856288419909149021807326666736635804532328355906924610206256225623256327737141846214373098833200106
26
22
$w_k$:
1.770106337397356352685723437869194702907400038089468175830646464558217617515929752102371808072129998e-7
26
23
$x_k$:
4.379602662983304971912542923699548675773315517468232635894148800810072768969364326989929465157323562
26
23
$w_k$:
2.524494034490533687075781476716255502286458470231026099265696894794611332260121675548250040482854483e-9
26
24
$x_k$:
4.941324957241379073791445944818101470023455076116482353598476391395138404222856114191833459333242791
26
24
$w_k$:
1.460999933981612369422693926806920339077808228657174780423914522514881529343845557636000587655407090e-11
26
25
$x_k$:
5.564524981950103097163064482378535941615149081748118204112946148628905909232343383432040025192071984
26
25
$w_k$:
2.383148659372160648946267218611284245698599016286779271153795050378592182605060113413992463313679339e-14
26
26
$x_k$:
6.309550385625693500904546656983340584766600795725299795284066690924610828355619393478832132373439161
26
26
$w_k$:
4.396916094753901864568217063627761319905514268245765309184922267549846248109151510157237666697736972e-18
27
1
$x_k$:
-6.451940140753471968598177229570380041282755394293097603365985253658025949585651176984044094692620663
27
1
$w_k$:
7.095779297051346812028789709326080647097188181851162078614206341772003387147969790044089724087560297e-19
27
2
$x_k$:
-5.712255552816536654428038957277432309287720833046377730372088004503578509765094774051778401657077720
27
2
$w_k$:
4.470772457393124628570908805064602274289539857040672445273450627713528432148061944846735834394123026e-15
27
3
$x_k$:
-5.093910003113183892145730282113367550493884391725904874737947283698815375644085067596741910425716225
27
3
$w_k$:
3.134117613623037753239014045812349563954648321419254560036085640386583218781063193358022764146469048e-12
27
4
$x_k$:
-4.536906663372441751241832377923223414159036738861051070539574806468281095721278103398694713315767106
27
4
$w_k$:
6.155031578231790658445648534586501391139805511394156066025823209578308888888144162718948836144495498e-10
27
5
$x_k$:
-4.018318670408738918532592176525327081298785886688483222428747934407040237828579934571582166883208456
27
5
$w_k$:
4.895400409699549319972390886295668297264689063777897248828071851623551537287392711348254560610970466e-8
27
6
$x_k$:
-3.526275340134353178367346927200604604029225756037375085200706452005940026676214071035820155046660100
27
6
$w_k$:
0.000001915280900595295665452571843650473704398728240889689159346466588033144944788902212099246327515124861
27
7
$x_k$:
-3.053582419822255277482232439048674657905337470405545060343762273675932534018771153486962055647556283
27
7
$w_k$:
0.00004146758004384136867496759165290748269330915592245669615268968900203262204164161138185102132010306225
27
8
$x_k$:
-2.595416338910818261197141210221972677731694947723231094220915420143439839511692029687180758448310911
27
8
$w_k$:
0.0005367696156881118797860901920047149253127016881880147937748395864279541326936086926850999172114063564
27
9
$x_k$:
-2.148296645361627364683113562294661512517626273815912112919266719584563436075476623696920296919681973
27
9
$w_k$:
0.004381279835792542265350549146967735971707709183557383827423093999929224233431715226086002304349106865
27
10
$x_k$:
-1.709560739260337350275400751697203755806908692961193954888476691963155968200970609735027615508602139
27
10
$w_k$:
0.02341593362534191185518713229805832525879702216435453627189274510033198897739278328015284427362984259
27
11
$x_k$:
-1.277066817339857988179321337744797475488187271639696695383653762204615904373789183479882466674934427
27
11
$w_k$:
0.08417308108405195350358304912253787763581392563077469899444505012642431719323366314918984080196595467
27
12
$x_k$:
-0.8490113420601030622824552933099780819658234155852674408297772860641688099453726843185921974030735821
27
12
$w_k$:
0.2073704807510094411417225940108476333694587248505584836257619783313484735281646579226685373759336795
27
13
$x_k$:
-0.4238079005438530410719988648029615493642324281969477376423352075714360034852438508210594783025906969
27
13
$w_k$:
0.3545173040997540217804596819310406308040596771633472043182193345590244995270100099321724237902471069
27
14
$x_k$:
0
equals: Zero
comment: $x_{14}=0$, the central node of every rule of odd order
27
14
$w_k$:
0.4235772880150595012886471973082862658087955252730834088013832175329614295816440585675430998274945588
27
15
$x_k$:
0.4238079005438530410719988648029615493642324281969477376423352075714360034852438508210594783025906969
27
15
$w_k$:
0.3545173040997540217804596819310406308040596771633472043182193345590244995270100099321724237902471069
27
16
$x_k$:
0.8490113420601030622824552933099780819658234155852674408297772860641688099453726843185921974030735821
27
16
$w_k$:
0.2073704807510094411417225940108476333694587248505584836257619783313484735281646579226685373759336795
27
17
$x_k$:
1.277066817339857988179321337744797475488187271639696695383653762204615904373789183479882466674934427
27
17
$w_k$:
0.08417308108405195350358304912253787763581392563077469899444505012642431719323366314918984080196595467
27
18
$x_k$:
1.709560739260337350275400751697203755806908692961193954888476691963155968200970609735027615508602139
27
18
$w_k$:
0.02341593362534191185518713229805832525879702216435453627189274510033198897739278328015284427362984259
27
19
$x_k$:
2.148296645361627364683113562294661512517626273815912112919266719584563436075476623696920296919681973
27
19
$w_k$:
0.004381279835792542265350549146967735971707709183557383827423093999929224233431715226086002304349106865
27
20
$x_k$:
2.595416338910818261197141210221972677731694947723231094220915420143439839511692029687180758448310911
27
20
$w_k$:
0.0005367696156881118797860901920047149253127016881880147937748395864279541326936086926850999172114063564
27
21
$x_k$:
3.053582419822255277482232439048674657905337470405545060343762273675932534018771153486962055647556283
27
21
$w_k$:
0.00004146758004384136867496759165290748269330915592245669615268968900203262204164161138185102132010306225
27
22
$x_k$:
3.526275340134353178367346927200604604029225756037375085200706452005940026676214071035820155046660100
27
22
$w_k$:
0.000001915280900595295665452571843650473704398728240889689159346466588033144944788902212099246327515124861
27
23
$x_k$:
4.018318670408738918532592176525327081298785886688483222428747934407040237828579934571582166883208456
27
23
$w_k$:
4.895400409699549319972390886295668297264689063777897248828071851623551537287392711348254560610970466e-8
27
24
$x_k$:
4.536906663372441751241832377923223414159036738861051070539574806468281095721278103398694713315767106
27
24
$w_k$:
6.155031578231790658445648534586501391139805511394156066025823209578308888888144162718948836144495498e-10
27
25
$x_k$:
5.093910003113183892145730282113367550493884391725904874737947283698815375644085067596741910425716225
27
25
$w_k$:
3.134117613623037753239014045812349563954648321419254560036085640386583218781063193358022764146469048e-12
27
26
$x_k$:
5.712255552816536654428038957277432309287720833046377730372088004503578509765094774051778401657077720
27
26
$w_k$:
4.470772457393124628570908805064602274289539857040672445273450627713528432148061944846735834394123026e-15
27
27
$x_k$:
6.451940140753471968598177229570380041282755394293097603365985253658025949585651176984044094692620663
27
27
$w_k$:
7.095779297051346812028789709326080647097188181851162078614206341772003387147969790044089724087560297e-19
28
1
$x_k$:
-6.591605442367742544012785215001567297873272605915479886516703208232957485739597490778749095304509462
28
1
$w_k$:
1.140139347903676163811173006224523018682204855997990501049280590001256021896329947371359903614795613e-19
28
2
$x_k$:
-5.857014641382850644987752330175007451395844339899988680980138049394383366665063957718116181235714061
28
2
$w_k$:
8.315937951206829881820500805669481626244408500581272906092603104520027541398528166989246845257242528e-16
28
3
$x_k$:
-5.243285373202935981516075399547689247660286995937579322579934273851789919249820170413367385460578911
28
3
$w_k$:
6.639436714909663635813411130976436644782071147938060042837234511551939578800839959297764752131982165e-13
28
4
$x_k$:
-4.690756523943117925360272134404862216329264930758481163799589625030061851672739491898060532420337247
28
4
$w_k$:
1.475853168277688614155519212651194976695747188067994352452279140128627627795605823840981805275485824e-10
28
5
$x_k$:
-4.176636742129268359391700276484577826084294579125539863981776796078505362835199191686935174456035358
28
5
$w_k$:
1.325682501541707815237201252423200243285887959464487544886270574564160060391649849203850271923565210e-8
28
6
$x_k$:
-3.689134238461679490758812336425523764943359059301534584056356898741895993776175911154798457958172750
28
6
$w_k$:
5.857719720992981734034979665454019674169504384459399659468536744999957484935874878743679068836143333e-7
28
7
$x_k$:
-3.221112076561455547601291432051913994437729133470206965563584789501206005254102923835304116177950903
28
7
$w_k$:
0.00001434550422971441531325893002041393674457694658275643353442403659318445994492842102971943979500083757
28
8
$x_k$:
-2.767795352913593763262473528989119023015910344217624133601651974886724591363420486488503011745473692
28
8
$w_k$:
0.0002106181000240323309183750307168382947787829520704002929870678528364415254166491954984925853624218411
28
9
$x_k$:
-2.325749842656441021771206134276337305085219682271539097303514762726226893832666357831298524059104970
28
9
$w_k$:
0.001957331294408989615675129121332762423870898190192217935271453531117065407069112759644717489182843999
28
10
$x_k$:
-1.892360496837685345971275944256557611692970786675570671952060344969422990849206000209094256384742760
28
10
$w_k$:
0.01196842321435481878660139167800889076328383474685012660627064261926656207830287643954909657117507616
28
11
$x_k$:
-1.465537263457409189606478321027745074635985900992006934007728615767287781216036482045530529235604228
28
11
$w_k$:
0.04951488928989813987814321814749261120845897254002495254874937892501189143128518202439845833863722685
28
12
$x_k$:
-1.043535273754208274794804541236774963752024639030146939765393866542887160883108982061343834960240314
28
12
$w_k$:
0.1413946097869549307037781514625205694820736617347645693313891027979872147384146790062808685480112889
28
13
$x_k$:
-0.6248367195052092274062469925293039920072250927503103865499636714995969718165177507009690956281382709
28
13
$w_k$:
0.2825613912593887351176106337856271255264107241954753189470937672405946959432229594190594782435943734
28
14
$x_k$:
-0.2080673826907368691410043575842291589696184192524250994331550823911811211754541555419489837266469608
28
14
$w_k$:
0.3986047178264514458787231443923852493197684180412473291961028772300670442452541381777077223246268472
28
15
$x_k$:
0.2080673826907368691410043575842291589696184192524250994331550823911811211754541555419489837266469608
28
15
$w_k$:
0.3986047178264514458787231443923852493197684180412473291961028772300670442452541381777077223246268472
28
16
$x_k$:
0.6248367195052092274062469925293039920072250927503103865499636714995969718165177507009690956281382709
28
16
$w_k$:
0.2825613912593887351176106337856271255264107241954753189470937672405946959432229594190594782435943734
28
17
$x_k$:
1.043535273754208274794804541236774963752024639030146939765393866542887160883108982061343834960240314
28
17
$w_k$:
0.1413946097869549307037781514625205694820736617347645693313891027979872147384146790062808685480112889
28
18
$x_k$:
1.465537263457409189606478321027745074635985900992006934007728615767287781216036482045530529235604228
28
18
$w_k$:
0.04951488928989813987814321814749261120845897254002495254874937892501189143128518202439845833863722685
28
19
$x_k$:
1.892360496837685345971275944256557611692970786675570671952060344969422990849206000209094256384742760
28
19
$w_k$:
0.01196842321435481878660139167800889076328383474685012660627064261926656207830287643954909657117507616
28
20
$x_k$:
2.325749842656441021771206134276337305085219682271539097303514762726226893832666357831298524059104970
28
20
$w_k$:
0.001957331294408989615675129121332762423870898190192217935271453531117065407069112759644717489182843999
28
21
$x_k$:
2.767795352913593763262473528989119023015910344217624133601651974886724591363420486488503011745473692
28
21
$w_k$:
0.0002106181000240323309183750307168382947787829520704002929870678528364415254166491954984925853624218411
28
22
$x_k$:
3.221112076561455547601291432051913994437729133470206965563584789501206005254102923835304116177950903
28
22
$w_k$:
0.00001434550422971441531325893002041393674457694658275643353442403659318445994492842102971943979500083757
28
23
$x_k$:
3.689134238461679490758812336425523764943359059301534584056356898741895993776175911154798457958172750
28
23
$w_k$:
5.857719720992981734034979665454019674169504384459399659468536744999957484935874878743679068836143333e-7
28
24
$x_k$:
4.176636742129268359391700276484577826084294579125539863981776796078505362835199191686935174456035358
28
24
$w_k$:
1.325682501541707815237201252423200243285887959464487544886270574564160060391649849203850271923565210e-8
28
25
$x_k$:
4.690756523943117925360272134404862216329264930758481163799589625030061851672739491898060532420337247
28
25
$w_k$:
1.475853168277688614155519212651194976695747188067994352452279140128627627795605823840981805275485824e-10
28
26
$x_k$:
5.243285373202935981516075399547689247660286995937579322579934273851789919249820170413367385460578911
28
26
$w_k$:
6.639436714909663635813411130976436644782071147938060042837234511551939578800839959297764752131982165e-13
28
27
$x_k$:
5.857014641382850644987752330175007451395844339899988680980138049394383366665063957718116181235714061
28
27
$w_k$:
8.315937951206829881820500805669481626244408500581272906092603104520027541398528166989246845257242528e-16
28
28
$x_k$:
6.591605442367742544012785215001567297873272605915479886516703208232957485739597490778749095304509462
28
28
$w_k$:
1.140139347903676163811173006224523018682204855997990501049280590001256021896329947371359903614795613e-19
29
1
$x_k$:
-6.728695198608849962822750901134364875494603637940179972337350728472061303029150529657380417815809351
29
1
$w_k$:
1.824460852767245336349913451710917031228099201753281003728404981484788700893914200521561306050058862e-20
29
2
$x_k$:
-5.998971289463820210647038871993437295779659726915946079522475474972951207795081561275483485697109294
29
2
$w_k$:
1.534500444605320849167987710826907875822802395531535303094692759467267800057853194721189133565140411e-16
29
3
$x_k$:
-5.389640521966751653742242061998290599094509811828410589842533882576533501898178436594588653384156090
29
3
$w_k$:
1.390107271449598028861994423061702828443007635277126877179345952041955427167521180697565199549119281e-13
29
4
$x_k$:
-4.841363651059164067515635198171676308015729830753973988494542334024763188002601373137383832050570923
29
4
$w_k$:
3.484130161308433998447025643921278058459085439768340610568483089834122514938907510304293913977191582e-11
29
5
$x_k$:
-4.331478293819150392192046072388661476065942056522986265387341655079913286476012750689752751864144129
29
5
$w_k$:
3.520312327600698362932787269841739691663900583227713233337623498283044255831888372956434650347086104e-9
29
6
$x_k$:
-3.848266792213620080200023467836908112391281036942890560382965830206341099702126547526012209159412848
29
6
$w_k$:
1.749229129949948357679594870640896620086184763397882151415713799030634919143173852054698981380242953e-7
29
7
$x_k$:
-3.384645141092214074937257479589172792998323486086082358709128445655502500303351258413329815395542379
29
7
$w_k$:
0.000004823073497647770212641905397090416501243902296798990296260277468998883640145457953478510772698110962
29
8
$x_k$:
-2.935882504290126484855959598429673091478643887726058895416131494576918787998347375974047835104138914
29
8
$w_k$:
0.00007990920354521804915959819511225498923055993898510477417587494255466873789639325689700784513967445291
29
9
$x_k$:
-2.498585691019404242566549210013830166441427430064480718972726231909261604359144449960399924289542586
29
9
$w_k$:
0.0008407925061402626159991853790836967355206170238468064454025436388358714444521986256874948051781709427
29
10
$x_k$:
-2.070181076053427740800217496483216578769987919000773354521351230706769463562244847028795576828836451
29
10
$w_k$:
0.005845503545271509647758321425692357384490532265272482183455567836231018301502837397359204538347236467
29
11
$x_k$:
-1.648622913892316411410947485807441090880798465261524268346394478395405653039974239501711572949217130
29
11
$w_k$:
0.02763965559202368832634302549650418131990429817834172251992176322672197830472357011800714831524030278
29
12
$x_k$:
-1.232215755084752903808592254564669893511847712247067614197044388217777748507188799341264422516397010
29
12
$w_k$:
0.09076884221557815976862447325830179350963186565780059659435358543234884298797506289191241518138565345
29
13
$x_k$:
-0.8194986812709115792987011701146148662765958134781160613646686236939049963666547345609181262566228865
29
13
$w_k$:
0.2101426944492106210349734723199570702270712736323578793705874608062690485463111144753995901921726930
29
14
$x_k$:
-0.4091646363949287367783019036997655186654411380616345571654378631543314657642679369970191772403608814
29
14
$w_k$:
0.3464189390716701863753722218023081343818149396624051451066589017731952385995105321176101940708861563
29
15
$x_k$:
0
equals: Zero
comment: $x_{15}=0$, the central node of every rule of odd order
29
15
$w_k$:
0.4089711746352298633131766042976557049188370588843563947047837962387213802857252979272829929368912981
29
16
$x_k$:
0.4091646363949287367783019036997655186654411380616345571654378631543314657642679369970191772403608814
29
16
$w_k$:
0.3464189390716701863753722218023081343818149396624051451066589017731952385995105321176101940708861563
29
17
$x_k$:
0.8194986812709115792987011701146148662765958134781160613646686236939049963666547345609181262566228865
29
17
$w_k$:
0.2101426944492106210349734723199570702270712736323578793705874608062690485463111144753995901921726930
29
18
$x_k$:
1.232215755084752903808592254564669893511847712247067614197044388217777748507188799341264422516397010
29
18
$w_k$:
0.09076884221557815976862447325830179350963186565780059659435358543234884298797506289191241518138565345
29
19
$x_k$:
1.648622913892316411410947485807441090880798465261524268346394478395405653039974239501711572949217130
29
19
$w_k$:
0.02763965559202368832634302549650418131990429817834172251992176322672197830472357011800714831524030278
29
20
$x_k$:
2.070181076053427740800217496483216578769987919000773354521351230706769463562244847028795576828836451
29
20
$w_k$:
0.005845503545271509647758321425692357384490532265272482183455567836231018301502837397359204538347236467
29
21
$x_k$:
2.498585691019404242566549210013830166441427430064480718972726231909261604359144449960399924289542586
29
21
$w_k$:
0.0008407925061402626159991853790836967355206170238468064454025436388358714444521986256874948051781709427
29
22
$x_k$:
2.935882504290126484855959598429673091478643887726058895416131494576918787998347375974047835104138914
29
22
$w_k$:
0.00007990920354521804915959819511225498923055993898510477417587494255466873789639325689700784513967445291
29
23
$x_k$:
3.384645141092214074937257479589172792998323486086082358709128445655502500303351258413329815395542379
29
23
$w_k$:
0.000004823073497647770212641905397090416501243902296798990296260277468998883640145457953478510772698110962
29
24
$x_k$:
3.848266792213620080200023467836908112391281036942890560382965830206341099702126547526012209159412848
29
24
$w_k$:
1.749229129949948357679594870640896620086184763397882151415713799030634919143173852054698981380242953e-7
29
25
$x_k$:
4.331478293819150392192046072388661476065942056522986265387341655079913286476012750689752751864144129
29
25
$w_k$:
3.520312327600698362932787269841739691663900583227713233337623498283044255831888372956434650347086104e-9
29
26
$x_k$:
4.841363651059164067515635198171676308015729830753973988494542334024763188002601373137383832050570923
29
26
$w_k$:
3.484130161308433998447025643921278058459085439768340610568483089834122514938907510304293913977191582e-11
29
27
$x_k$:
5.389640521966751653742242061998290599094509811828410589842533882576533501898178436594588653384156090
29
27
$w_k$:
1.390107271449598028861994423061702828443007635277126877179345952041955427167521180697565199549119281e-13
29
28
$x_k$:
5.998971289463820210647038871993437295779659726915946079522475474972951207795081561275483485697109294
29
28
$w_k$:
1.534500444605320849167987710826907875822802395531535303094692759467267800057853194721189133565140411e-16
29
29
$x_k$:
6.728695198608849962822750901134364875494603637940179972337350728472061303029150529657380417815809351
29
29
$w_k$:
1.824460852767245336349913451710917031228099201753281003728404981484788700893914200521561306050058862e-20
30
1
$x_k$:
-6.863345293529891581061108357555026621483711048091473191109438239998187181595786970915569936308590927
30
1
$w_k$:
2.908254700131226229411027473651280300111142807523348483692341919540039336368974931241400435284145056e-21
30
2
$x_k$:
-6.138279220123934620394992378537579501065097690098971665763418042960451552797917599555529066225031762
30
2
$w_k$:
2.810333602750903708762774915339798584733124198786098042754862186296199564290341573124443870233410870e-17
30
3
$x_k$:
-5.533147151567495725118333555580396733256258281089916942497248659086904759314596675185634704419094060
30
3
$w_k$:
2.878607080548706062192397911415697681194809630083876745136651958298644850067947857914807428941749890e-14
30
4
$x_k$:
-4.988918968589943944486497106330954295380955882987156842849732696976732054427048940653837477254772772
30
4
$w_k$:
8.106186297463044203993447961730650005250064689842702563516955778566906953216668283464338910627818567e-12
30
5
$x_k$:
-4.483055357092518341887037619709105216099539530886469812908056682520065442934523808366224509891844985
30
5
$w_k$:
9.178580424378528208500757424924177490250224373331011652833525764825202491341015670892373027516794845e-10
30
6
$x_k$:
-4.003908603861228815227876013321818076299208459902075297492767592901869993664614970088641318140913678
30
6
$w_k$:
5.108522450775946277389632044028376898650679853092962128866807731124967357634807005506301943239715848e-8
30
7
$x_k$:
-3.544443873155349886925400902168363649838349904055592526215203322768846952527595959016152901700849998
30
7
$w_k$:
0.000001579094887324710288346387940218913608500163765690317806912096412848325425348556227535504401465140957
30
8
$x_k$:
-3.099970529586441748688733322374639019330853820344286370205981120228570160768389176579052818543663268
30
8
$w_k$:
0.00002938725228922987641501184234118690589133227323379648691305935023027249765129793451271275701335539181
30
9
$x_k$:
-2.667132124535617200571106464220874917419224635112664777394138974103006747688331139147888126251952570
30
9
$w_k$:
0.0003483101243186855234209953231825356603725734535520025063737149080802193251576304282655672483109797881
30
10
$x_k$:
-2.243391467761504072472979994825061361769907205754983901927009210119383620988687223446003600100492594
30
10
$w_k$:
0.002737922473067658462989425689526626378714893886720025763709441150581201188773912380571480453045185940
30
11
$x_k$:
-1.826741143603688038835880483506128146129689331638131667194707668356058155845779212751021363820723162
30
11
$w_k$:
0.01470382970482668351527735577870225653132696540980725066521141913249943222665823343458835604241606872
30
12
$x_k$:
-1.415527800198188511940725105547579822528919868313258053947391397314291998562062429865333013515738857
30
12
$w_k$:
0.05514417687023425116807549481830163096515716211602690997890176976416378565806634946562828044714816839
30
13
$x_k$:
-1.008338271046723461804989608696417905847615791082148642552323570751489375343851618045258800715220112
30
13
$w_k$:
0.1467358475408900997516936431518417217412648095362947961541268996868414299272096262557791296575085889
30
14
$x_k$:
-0.6039210586255523077781556787573418169977445699802587059532106162656622990092529832182396425244224903
30
14
$w_k$:
0.2801309308392126674134932112934340820472815711769870011142263913657513255437848702874063784554483561
30
15
$x_k$:
-0.2011285765488714855457630132436921856934107774537679022921630106054280758859968531105506161739015741
30
15
$w_k$:
0.3863948895418138625556018491651224636809651966592859824394151615209931415401308981790712356498952361
30
16
$x_k$:
0.2011285765488714855457630132436921856934107774537679022921630106054280758859968531105506161739015741
30
16
$w_k$:
0.3863948895418138625556018491651224636809651966592859824394151615209931415401308981790712356498952361
30
17
$x_k$:
0.6039210586255523077781556787573418169977445699802587059532106162656622990092529832182396425244224903
30
17
$w_k$:
0.2801309308392126674134932112934340820472815711769870011142263913657513255437848702874063784554483561
30
18
$x_k$:
1.008338271046723461804989608696417905847615791082148642552323570751489375343851618045258800715220112
30
18
$w_k$:
0.1467358475408900997516936431518417217412648095362947961541268996868414299272096262557791296575085889
30
19
$x_k$:
1.415527800198188511940725105547579822528919868313258053947391397314291998562062429865333013515738857
30
19
$w_k$:
0.05514417687023425116807549481830163096515716211602690997890176976416378565806634946562828044714816839
30
20
$x_k$:
1.826741143603688038835880483506128146129689331638131667194707668356058155845779212751021363820723162
30
20
$w_k$:
0.01470382970482668351527735577870225653132696540980725066521141913249943222665823343458835604241606872
30
21
$x_k$:
2.243391467761504072472979994825061361769907205754983901927009210119383620988687223446003600100492594
30
21
$w_k$:
0.002737922473067658462989425689526626378714893886720025763709441150581201188773912380571480453045185940
30
22
$x_k$:
2.667132124535617200571106464220874917419224635112664777394138974103006747688331139147888126251952570
30
22
$w_k$:
0.0003483101243186855234209953231825356603725734535520025063737149080802193251576304282655672483109797881
30
23
$x_k$:
3.099970529586441748688733322374639019330853820344286370205981120228570160768389176579052818543663268
30
23
$w_k$:
0.00002938725228922987641501184234118690589133227323379648691305935023027249765129793451271275701335539181
30
24
$x_k$:
3.544443873155349886925400902168363649838349904055592526215203322768846952527595959016152901700849998
30
24
$w_k$:
0.000001579094887324710288346387940218913608500163765690317806912096412848325425348556227535504401465140957
30
25
$x_k$:
4.003908603861228815227876013321818076299208459902075297492767592901869993664614970088641318140913678
30
25
$w_k$:
5.108522450775946277389632044028376898650679853092962128866807731124967357634807005506301943239715848e-8
30
26
$x_k$:
4.483055357092518341887037619709105216099539530886469812908056682520065442934523808366224509891844985
30
26
$w_k$:
9.178580424378528208500757424924177490250224373331011652833525764825202491341015670892373027516794845e-10
30
27
$x_k$:
4.988918968589943944486497106330954295380955882987156842849732696976732054427048940653837477254772772
30
27
$w_k$:
8.106186297463044203993447961730650005250064689842702563516955778566906953216668283464338910627818567e-12
30
28
$x_k$:
5.533147151567495725118333555580396733256258281089916942497248659086904759314596675185634704419094060
30
28
$w_k$:
2.878607080548706062192397911415697681194809630083876745136651958298644850067947857914807428941749890e-14
30
29
$x_k$:
6.138279220123934620394992378537579501065097690098971665763418042960451552797917599555529066225031762
30
29
$w_k$:
2.810333602750903708762774915339798584733124198786098042754862186296199564290341573124443870233410870e-17
30
30
$x_k$:
6.863345293529891581061108357555026621483711048091473191109438239998187181595786970915569936308590927
30
30
$w_k$:
2.908254700131226229411027473651280300111142807523348483692341919540039336368974931241400435284145056e-21
Definition
For $n\geq 1$ the Gauss–Hermite quadrature rule with $n$ points is the unique rule $\int_{-\infty}^{\infty} f(x)\,e^{-x^2}\,\mathrm{d}x \approx \sum_{k=1}^{n} w_k f(x_k)$ that is exact for every polynomial $f$ of degree at most $2n-1$ [2]. Listed are its nodes $x_1<\cdots<x_n$ and its weights $w_k$.
Parameters
$n$
—   number of nodes ($n\geq 1$)
$k$
—   index of the node, in increasing order ($1\leq k\leq n$)
Formulas
(1)
$w_k=\dfrac{2^{n-1}\,n!\,\sqrt{\pi}}{n^2\,H_{n-1}(x_k)^2}=\dfrac{2^{n+1}\,n!\,\sqrt{\pi}}{H_n'(x_k)^2}$ [1], the two equal because $H_n'=2nH_{n-1}$. Also $\dfrac{1}{w_k}=\sum_{j=0}^{n-1}\dfrac{H_j(x_k)^2}{2^j\,j!\,\sqrt{\pi}}$, the Christoffel function at the node.
(2)
$\sum_{k=1}^{n} w_k x_k^m=\int_{-\infty}^{\infty}x^m e^{-x^2}\,\mathrm{d}x$, which is $\Gamma\bigl(\frac{m+1}{2}\bigr)=\sqrt{\pi}\,\dfrac{(m-1)!!}{2^{m/2}}$ for even $m$ and $0$ for odd $m$, for $0\leq m\leq 2n-1$, and not for $m=2n$.
(3)
$x_{n+1-k}=-x_k$, $w_{n+1-k}=w_k$, $w_k>0$ and $\sum_{k=1}^{n}w_k=\sqrt{\pi}$.
(4)
For odd $n$, $x_{(n+1)/2}=0$ and $w_{(n+1)/2}=\dfrac{2^{n+1}\,n!\,\sqrt{\pi}}{H_n'(0)^2} =\dfrac{2^{n-1}\,n!\,\sqrt{\pi}}{n^2\,H_{n-1}(0)^2}$ with $H_{n-1}(0)=(-1)^{(n-1)/2}\dfrac{(n-1)!}{((n-1)/2)!}$: $\sqrt{\pi}$, $\frac{2}{3}\sqrt{\pi}$, $\frac{8}{15}\sqrt{\pi}$, $\frac{16}{35}\sqrt{\pi}$, $\frac{128}{315}\sqrt{\pi}$ for $n=1,3,5,7,9$.
(5)
For the weight $e^{-x^2/2}$ the nodes are $\sqrt{2}\,x_k$, the roots of $He_n(x)=2^{-n/2}H_n(x/\sqrt{2})$, and the weights are $\sqrt{2}\,w_k$, summing to $\sqrt{2\pi}$.
(6)
For $Y$ normally distributed with mean $\mu$ and variance $\sigma^2$, $\mathbb{E}\,f(Y)=\dfrac{1}{\sigma\sqrt{2\pi}}\int_{-\infty}^{\infty} f(y)\,e^{-(y-\mu)^2/(2\sigma^2)}\,\mathrm{d}y\approx\dfrac{1}{\sqrt{\pi}} \sum_{k=1}^{n}w_k\,f\bigl(\sqrt{2}\,\sigma x_k+\mu\bigr)$ [2].
(7)
The nodes of the rules with $n$ and $n-1$ points interlace: $x_k^{(n)}<x_k^{(n-1)}<x_{k+1}^{(n)}$ for $1\leq k\leq n-1$, as the roots of consecutive orthogonal polynomials do.
Comments
(8)
The nodes are the roots of $H_n$, the Hermite polynomial in the physicists' convention, $H_n(x)=(-1)^n e^{x^2}\frac{d^n}{dx^n}e^{-x^2}$, orthogonal for the weight $e^{-x^2}$ on the real line. Its $n$ roots are real and simple, and $x_1<\cdots<x_n$ is their increasing order. Placing the nodes at these roots is what lets a rule with $n$ points reach degree $2n-1$; a generic choice of $n$ points gives exactly $n-1$.
(9)
The same rule for the weight $e^{-x^2/2}$, whose orthogonal polynomials are the Hermite polynomials $He_n$ in the probabilists' convention, has nodes $\sqrt{2}\,x_k$ and weights $\sqrt{2}\,w_k$, and is not listed separately. For a random variable $Z$ with the standard normal distribution the rule reads $\mathbb{E}\,f(Z)\approx\frac{1}{\sqrt{\pi}}\sum_{k=1}^{n}w_k f(\sqrt{2}\,x_k)$, which is how it is used to take expectations under a Gaussian.
(10)
Every rule is symmetric, $x_{n+1-k}=-x_k$ and $w_{n+1-k}=w_k$, and both halves are listed rather than the nonnegative one.
(11)
Every node is an algebraic number and every weight is $\sqrt{\pi}$ times an algebraic number. The node $x=0$ of every rule of odd order is written exactly; its weight $w_{(n+1)/2}=2^{n+1}n!\sqrt{\pi}/H_n'(0)^2$ is a rational multiple of $\sqrt{\pi}$, given in the comment on the entry for $n\leq 9$. The weight of the one-point rule is $\sqrt{\pi}=\Gamma(1/2)$ and the two weights of the two-point rule are $\sqrt{\pi}/2=\Gamma(3/2)$, both in the table of values of the Gamma function.
(12)
The rules with $n\leq 5$ have closed forms in square roots, given in the comments on their entries, and the nodes $\pm 1/\sqrt{2}$ and $\pm\sqrt{3/2}$ of $n=2,3$ are also in the table of quadratic algebraic numbers. The polynomials whose roots are the normalised weights $w_k/\sqrt{\pi}$ are OEIS A393904 [7].
(13)
The nodes are unbounded, the largest growing like $\sqrt{2n}$: at $n=30$ it is $6.8633\ldots$, and the weight there, $2.9\cdot 10^{-21}$, is small because $e^{-x^2}$ is.
(14)
Gauss–Legendre quadrature is the same construction for the weight $1$ on $[-1,1]$. The Gauss–Laguerre rule, for $e^{-x}$ on $[0,\infty)$, and the Gauss–Chebyshev rule, whose nodes $\cos\frac{(2k-1)\pi}{2n}$ are in the table of $\cos(\pi x)$ at rational $x$, are different tables.
Programs
(P1)
Sage
R.<x> = ZZ[]
RIF400 = RealIntervalField(400)
p = R(hermite(8, x))
nodes = p.roots(RIF400, multiplicities=False)   # x_1 = -2.93063742025724401922...
weights = [2^7*factorial(8)*RIF400(pi).sqrt()/(64*R(hermite(7, x))(r)^2) for r in nodes]   # w_1 = 0.000199604072211367619...
(P2)
Python
import numpy.polynomial.hermite as H
nodes, weights = H.hermgauss(8)      # double precision: -2.93063742, 1.99604072e-04
References
[1]
M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, Dover, 1972, §25.4.46 and Table 25.10.
Links
Similar tables
Hermite polynomials in physicist's convention —   the nodes are the roots of $H_n$
Hermite polynomials in probabilist's convention —   the rule for $e^{-x^2/2}$ has nodes $\sqrt{2}\,x_k$, the roots of $He_n$, and weights $\sqrt{2}\,w_k$
Nodes and weights of Gauss–Legendre quadrature —   the same construction for the weight $1$ on $[-1,1]$
Values of the Gamma function at rational numbers —   holds $\sqrt{\pi}=\Gamma(1/2)$ and $\sqrt{\pi}/2=\Gamma(3/2)$, the weights of $n=1,2$, and the moments $\Gamma(\frac{m+1}{2})$
Algebraic numbers of degree 2 —   holds the nodes $\pm 1/\sqrt{2}$ and $\pm\sqrt{3/2}$ of $n=2,3$
Data properties
Entries are of type: real number
How well the digits are known: proven
Table is complete: no (it holds every rule with $n\leq 30$, where $n$ is the number of nodes; each rule is listed in full, every node with its weight, and both halves of the symmetric set)
How they were obtained:

$H_n$ is built exactly in $\mathbb{Z}[x]$; its roots are isolated by Sage's real root isolation over the interval field, so each node is an interval provably containing one root, carried on as an arb ball at 397 bits; $w_k$ is $2^{n-1}n!\sqrt{\pi}/(n^2H_{n-1}(x_k)^2)$ in ball arithmetic, $\sqrt{\pi}$ included, and the digits written are those the ball supports (every entry supports more than 110).

more

Before a rule is written it must also satisfy the Christoffel formula for the weights, the symmetry, $\sum w_k=\sqrt{\pi}$, the central-weight formula, and exactness on $x^m$ for $m\leq 2n-1$ with failure at $m=2n$. Outside the generator the values were compared with the closed forms for $n\leq 5$, with the stored polynomials $H_n$ and $He_n$, with twenty-five OEIS expansions to 100 digits, with the OEIS A393904 weight polynomials for $n\leq 7$, and with DLMF Tables 3.5.10–3.5.13 ($n=5,10,15,20$), with the controls that must fail failing.



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