History of Boole polynomials $r_n(x)$

     the printed values of $s_0(x;1)$ through $s_2(x;1)$ with $r_n(x)=2s_n(x;1)$, against     CITE{formula-finite-difference} and CITE{formula-special-value} on every entry,-    and against CITE{formula-euler-transform} with the Euler polynomials computed-    from their derivative and reflection identities.+    and against CITE{formula-euler-transform} with the HREF{Euler_polynomials}[Euler+    polynomials] computed from their derivative and reflection identities. Display properties:   number-header: $r_n(x)$ Numbers:-- params:-    n: '0'-  number: '1'-- params:-    n: '1'-  number: x - 1/2-- params:-    n: '2'-  number: x^2 - 2*x + 1/2-- params:-    n: '3'-  number: x^3 - 9/2*x^2 + 5*x - 3/4-- params:-    n: '4'-  number: x^4 - 8*x^3 + 20*x^2 - 16*x + 3/2-- params:-    n: '5'-  number: x^5 - 25/2*x^4 + 55*x^3 - 100*x^2 + 64*x - 15/4-- params:-    n: '6'-  number: x^6 - 18*x^5 + 245/2*x^4 - 390*x^3 + 574*x^2 - 312*x + 45/4-- params:-    n: '7'-  number: x^7 - 49/2*x^6 + 238*x^5 - 4655/4*x^4 + 2989*x^3 - 3773*x^2 + 1812*x - 315/8-- params:-    n: '8'-  number: x^8 - 32*x^7 + 420*x^6 - 2912*x^5 + 11424*x^4 - 25088*x^3 + 28160*x^2 --    12288*x + 315/2-- params:-    n: '9'-  number: x^9 - 81/2*x^8 + 690*x^7 - 6426*x^6 + 35553*x^5 - 118692*x^4 + 231020*x^3-    - 236304*x^2 + 95616*x - 2835/4-- params:-    n: '10'-  number: x^10 - 50*x^9 + 2145/2*x^8 - 12900*x^7 + 95403*x^6 - 447090*x^5 + 1317140*x^4+  '0': '1'+  '1': x - 1/2+  '2': x^2 - 2*x + 1/2+  '3': x^3 - 9/2*x^2 + 5*x - 3/4+  '4': x^4 - 8*x^3 + 20*x^2 - 16*x + 3/2+  '5': x^5 - 25/2*x^4 + 55*x^3 - 100*x^2 + 64*x - 15/4+  '6': x^6 - 18*x^5 + 245/2*x^4 - 390*x^3 + 574*x^2 - 312*x + 45/4+  '7': x^7 - 49/2*x^6 + 238*x^5 - 4655/4*x^4 + 2989*x^3 - 3773*x^2 + 1812*x - 315/8+  '8': x^8 - 32*x^7 + 420*x^6 - 2912*x^5 + 11424*x^4 - 25088*x^3 + 28160*x^2 - 12288*x+    + 315/2+  '9': x^9 - 81/2*x^8 + 690*x^7 - 6426*x^6 + 35553*x^5 - 118692*x^4 + 231020*x^3 -+    236304*x^2 + 95616*x - 2835/4+  '10': x^10 - 50*x^9 + 2145/2*x^8 - 12900*x^7 + 95403*x^6 - 447090*x^5 + 1317140*x^4     - 2327800*x^3 + 2208096*x^2 - 840960*x + 14175/4-- params:-    n: '11'-  number: x^11 - 121/2*x^10 + 1595*x^9 - 96195/4*x^8 + 228723*x^7 - 2853543/2*x^6-    + 5875925*x^5 - 15653770*x^4 + 25556476*x^3 - 22773168*x^2 + 8254080*x - 155925/8-- params:-    n: '12'-  number: x^12 - 72*x^11 + 2288*x^10 - 42240*x^9 + 1003431/2*x^8 - 4009896*x^7 + 21900164*x^6+  '11': x^11 - 121/2*x^10 + 1595*x^9 - 96195/4*x^8 + 228723*x^7 - 2853543/2*x^6 ++    5875925*x^5 - 15653770*x^4 + 25556476*x^3 - 22773168*x^2 + 8254080*x - 155925/8+  '12': x^12 - 72*x^11 + 2288*x^10 - 42240*x^9 + 1003431/2*x^8 - 4009896*x^7 + 21900164*x^6     - 81251280*x^5 + 199180696*x^4 - 304256832*x^3 + 257182848*x^2 - 89441280*x +     467775/4-- params:-    n: '13'-  number: x^13 - 169/2*x^12 + 3185*x^11 - 70642*x^10 + 1024023*x^9 - 40751139/4*x^8+  '13': x^13 - 169/2*x^12 + 3185*x^11 - 70642*x^10 + 1024023*x^9 - 40751139/4*x^8     + 71054555*x^7 - 348421216*x^6 + 1185340156*x^5 - 2708689412*x^4 + 3909228960*x^3     - 3158131392*x^2 + 1060369920*x - 6081075/8-- params:-    n: '14'-  number: x^14 - 98*x^13 + 8645/2*x^12 - 113386*x^11 + 1968967*x^10 - 23837814*x^9-    + 825403865/4*x^8 - 1288325038*x^7 + 5775067298*x^6 - 18255084848*x^5 + 39274578980*x^4+  '14': x^14 - 98*x^13 + 8645/2*x^12 - 113386*x^11 + 1968967*x^10 - 23837814*x^9 ++    825403865/4*x^8 - 1288325038*x^7 + 5775067298*x^6 - 18255084848*x^5 + 39274578980*x^4     - 53961319776*x^3 + 41909678784*x^2 - 13649610240*x + 42567525/8-- params:-    n: '15'-  number: x^15 - 225/2*x^14 + 5740*x^13 - 702975/4*x^12 + 3600142*x^11 - 104159055/2*x^10+  '15': x^15 - 225/2*x^14 + 5740*x^13 - 702975/4*x^12 + 3600142*x^11 - 104159055/2*x^10     + 547195220*x^9 - 33832688175/8*x^8 + 24071760713*x^7 - 99976371495*x^6 + 296634742040*x^5     - 605548602750*x^4 + 796866696144*x^3 - 597788238240*x^2 + 189550368000*x - 638512875/16-- params:-    n: '16'-  number: x^16 - 128*x^15 + 7480*x^14 - 264320*x^13 + 6305572*x^12 - 107359616*x^11+  '16': x^16 - 128*x^15 + 7480*x^14 - 264320*x^13 + 6305572*x^12 - 107359616*x^11     + 1344731960*x^10 - 12585189760*x^9 + 88463817728*x^8 - 465377296384*x^7 + 1809483079040*x^6     - 5079891281920*x^5 + 9901384525824*x^4 - 12540751183872*x^3 + 9121468907520*x^2     - 2824077312000*x + 638512875/2-- params:-    n: '17'-  number: x^17 - 289/2*x^16 + 9588*x^15 - 387260*x^14 + 10640742*x^13 - 210549794*x^12+  '17': x^17 - 289/2*x^16 + 9588*x^15 - 387260*x^14 + 10640742*x^13 - 210549794*x^12     + 3097588156*x^11 - 34487381500*x^10 + 292927290513*x^9 - 1898843734216*x^8 +     9330230497224*x^7 - 34242173230720*x^6 + 91545085129744*x^5 - 171239517345408*x^4     + 209588629900032*x^3 - 148266768107520*x^2 + 44927447040000*x - 10854718875/4-- params:-    n: '18'-  number: x^18 - 162*x^17 + 24225/2*x^16 - 554472*x^15 + 17381922*x^14 - 395417484*x^13+  '18': x^18 - 162*x^17 + 24225/2*x^16 - 554472*x^15 + 17381922*x^14 - 395417484*x^13     + 6748170910*x^12 - 88080987384*x^11 + 888311328333*x^10 - 6944450916546*x^9 +     41961438905880*x^8 - 194200540659216*x^7 + 677192208310864*x^6 - 1733205672011808*x^5     + 3124469631836160*x^4 - 3707900113724928*x^3 + 2557806503546880*x^2 - 760034451456000*x     + 97692469875/4-- params:-    n: '19'-  number: x^19 - 361/2*x^18 + 15105*x^17 - 3111459/4*x^16 + 27591306*x^15 - 714917541*x^14+  '19': x^19 - 361/2*x^18 + 15105*x^17 - 3111459/4*x^16 + 27591306*x^15 - 714917541*x^14     + 14003403370*x^13 - 211668327377*x^12 + 2498342766621*x^11 - 46299422056173/2*x^10     + 168390024439845*x^9 - 956555351152908*x^8 + 4198030176812536*x^7 - 13984853571016232*x^6     + 34416166165033680*x^5 - 60003715510162944*x^4 + 69237300674209536*x^3 - 46676149842216960*x^2     + 13622700994560000*x - 1856156927625/8-- params:-    n: '20'-  number: x^20 - 200*x^19 + 18620*x^18 - 1071600*x^17 + 85391187/2*x^16 - 1249854960*x^15+  '20': x^20 - 200*x^19 + 18620*x^18 - 1071600*x^17 + 85391187/2*x^16 - 1249854960*x^15     + 27842109040*x^14 - 482286545600*x^13 + 6581910031151*x^12 - 71264075418120*x^11     + 613419165783060*x^10 - 4187758999866000*x^9 + 22519190501472976*x^8 - 94241205130638080*x^7     + 301278272240281280*x^6 - 715546448995564800*x^5 + 1210153230842121216*x^4 -     1360982737083248640*x^3 + 898326645239808000*x^2 - 257872110354432000*x + 9280784638125/4-- params:-    n: '21'-  number: x^21 - 441/2*x^20 + 22715*x^19 - 1452360*x^18 + 64579746*x^17 - 8482338207/4*x^16+  '21': x^21 - 441/2*x^20 + 22715*x^19 - 1452360*x^18 + 64579746*x^17 - 8482338207/4*x^16     + 53295248710*x^15 - 1048453329420*x^14 + 16374285724181*x^13 - 409390476453231/2*x^12     + 2055807802430655*x^11 - 16583201106233580*x^10 + 107002281597887896*x^9 - 547785143426856888*x^8

     - 25577540168993261568*x^4 + 28094078493014814720*x^3 - 18185377811779584000*x^2     + 5140559166898176000*x - 194896477400625/8-- params:-    n: '22'-  number: x^22 - 242*x^21 + 54901/2*x^20 - 1939630*x^19 + 95697756*x^18 - 3502544892*x^17+  '22': x^22 - 242*x^21 + 54901/2*x^20 - 1939630*x^19 + 95697756*x^18 - 3502544892*x^17     + 394464395677/4*x^16 - 2185966124540*x^15 + 38721598493501*x^14 - 553218142768522*x^13     + 12812942943846591/2*x^12 - 60214420913596950*x^11 + 458434321443604726*x^10

     - 607666766286379345920*x^3 + 386283966709745664000*x^2 - 107637093007589376000*x     + 2143861251406875/8-- params:-    n: '23'-  number: x^23 - 529/2*x^22 + 32890*x^21 - 10223983/4*x^20 + 139202371*x^19 - 5646571392*x^18+  '23': x^23 - 529/2*x^22 + 32890*x^21 - 10223983/4*x^20 + 139202371*x^19 - 5646571392*x^18     + 176996624200*x^17 - 35121409928011/8*x^16 + 87521026854151*x^15 - 2833097687230349/2*x^14     + 18725054488924210*x^13 - 810724326981796773/4*x^12 + 1795696721692314661*x^11

     - 13051316816951548177152*x^4 + 13744314486064293166080*x^3 - 8590742396497529856000*x^2     + 2361827297364885504000*x - 49308808782358125/16-- params:-    n: '24'-  number: x^24 - 288*x^23 + 39100*x^22 - 3327456*x^21 + 199095820*x^20 - 8905098048*x^19+  '24': x^24 - 288*x^23 + 39100*x^22 - 3327456*x^21 + 199095820*x^20 - 8905098048*x^19     + 309035300200*x^18 - 8524549826496*x^17 + 379909253385695/2*x^16 - 3456298360894368*x^15     + 51700392572086300*x^14 - 638057368168404576*x^13 + 6502557037952959840*x^12

     - 205050245511374911143936*x^5 + 313991700089360088858624*x^4 - 324471624108838378536960*x^3     + 199627875410463424512000*x^2 - 54193944307263602688000*x + 147926426347074375/4-- params:-    n: '25'-  number: x^25 - 625/2*x^24 + 46150*x^23 - 4283750*x^22 + 280403695*x^21 - 13765540250*x^20+  '25': x^25 - 625/2*x^24 + 46150*x^23 - 4283750*x^22 + 280403695*x^21 - 13765540250*x^20     + 526222239400*x^19 - 16054166232500*x^18 + 397443552698335*x^17 - 32300076622100375/4*x^16     + 135650640887353150*x^15 - 1892454976221293750*x^14 + 21966662302344164065*x^13

     + 7981390674638577451008000*x^3 - 4838135659029511372800000*x^2 + 1297872705574034472960000*x     - 3698160658676859375/8-- params:-    n: '26'-  number: x^26 - 338*x^25 + 108225/2*x^24 - 5458700*x^23 + 389374245*x^22 - 20892352910*x^21+  '26': x^26 - 338*x^25 + 108225/2*x^24 - 5458700*x^23 + 389374245*x^22 - 20892352910*x^21     + 875781599550*x^20 - 29404826937200*x^19 + 804371465389635*x^18 - 18139519503621230*x^17     + 1359747273778449075/4*x^16 - 5320831185010144700*x^15 + 69747861617871300615*x^14

     + 204538480185068310647808000*x^4 - 204238250318652668411904000*x^3 + 122085892379084851445760000*x^2     - 32383555215793434132480000*x + 48076088562799171875/8-- params:-    n: '27'-  number: x^27 - 729/2*x^26 + 63063*x^25 - 27562275/4*x^24 + 533705445*x^23 - 62358960105/2*x^22+  '27': x^27 - 729/2*x^26 + 63063*x^25 - 27562275/4*x^24 + 533705445*x^23 - 62358960105/2*x^22     + 1427301067335*x^21 - 52504558402725*x^20 + 1579294851469335*x^19 - 78638236029696915/2*x^18     + 817136669003787405*x^17 - 114044165659754053425/8*x^16 + 209468862114969832815*x^15

     + 3821599085879018504848499712*x^5 - 5522577449692134922781184000*x^4 + 5428890968370642427428864000*x^3     - 3202614106265207774085120000*x^2 + 840469456539816996372480000*x - 1298054391195577640625/16-- params:-    n: '28'-  number: x^28 - 392*x^27 + 73080*x^26 - 8622432*x^25 + 1445604615/2*x^24 - 45815154840*x^23+  '28': x^28 - 392*x^27 + 73080*x^26 - 8622432*x^25 + 1445604615/2*x^24 - 45815154840*x^23     + 2281686072900*x^22 - 91585587933840*x^21 + 3015794189292885*x^20 - 82493132723721720*x^19     + 1891143595179165600*x^18 - 36557122228552336320*x^17 + 2392828079202044924835/4*x^16

     - 149694142021195004168011776000*x^3 + 87210162045823696021094400000*x^2 - 22655441841975790109982720000*x     + 9086380738369043484375/8-- params:-    n: '29'-  number: x^29 - 841/2*x^28 + 84245*x^27 - 10702566*x^26 + 968067009*x^25 - 265445094915/4*x^24+  '29': x^29 - 841/2*x^28 + 84245*x^27 - 10702566*x^26 + 968067009*x^25 - 265445094915/4*x^24     + 3583104898425*x^23 - 156352674908820*x^22 + 5613615840447615*x^21 - 335944941908460885/2*x^20     + 4227550501953480975*x^19 - 90077717395793255310*x^18 + 1631989850359931268555*x^17

     - 4476919953141720090741537792000*x^4 + 4276249340857606632772288512000*x^3 -     2461896026741963985615912960000*x^2 + 633392251320362817096253440000*x - 263505041412702261046875/16-- params:-    n: '30'-  number: x^30 - 450*x^29 + 193285/2*x^28 - 13184850*x^27 + 1283224509*x^26 - 94849856010*x^25+  '30': x^30 - 450*x^29 + 193285/2*x^28 - 13184850*x^27 + 1283224509*x^26 - 94849856010*x^25     + 22138971308025/4*x^24 - 261659569772250*x^23 + 10205693517740565*x^22 - 332730812462998950*x^21     + 18308094685695994725/2*x^20 - 213979995041323533750*x^19 + 4270105261687986490755*x^18