Euler polynomials $E_n$
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Numbers
$n$ 
0:
1
1:
x - 1/2
2:
x^2 - x
3:
x^3 - 3/2*x^2 + 1/4
4:
x^4 - 2*x^3 + x
5:
x^5 - 5/2*x^4 + 5/2*x^2 - 1/2
6:
x^6 - 3*x^5 + 5*x^3 - 3*x
7:
x^7 - 7/2*x^6 + 35/4*x^4 - 21/2*x^2 + 17/8
8:
x^8 - 4*x^7 + 14*x^5 - 28*x^3 + 17*x
9:
x^9 - 9/2*x^8 + 21*x^6 - 63*x^4 + 153/2*x^2 - 31/2
10:
x^10 - 5*x^9 + 30*x^7 - 126*x^5 + 255*x^3 - 155*x
11:
x^11 - 11/2*x^10 + 165/4*x^8 - 231*x^6 + 2805/4*x^4 - 1705/2*x^2 + 691/4
12:
x^12 - 6*x^11 + 55*x^9 - 396*x^7 + 1683*x^5 - 3410*x^3 + 2073*x
13:
x^13 - 13/2*x^12 + 143/2*x^10 - 1287/2*x^8 + 7293/2*x^6 - 22165/2*x^4 + 26949/2*x^2 - 5461/2
14:
x^14 - 7*x^13 + 91*x^11 - 1001*x^9 + 7293*x^7 - 31031*x^5 + 62881*x^3 - 38227*x
15:
x^15 - 15/2*x^14 + 455/4*x^12 - 3003/2*x^10 + 109395/8*x^8 - 155155/2*x^6 + 943215/4*x^4 - 573405/2*x^2 + 929569/16
16:
x^16 - 8*x^15 + 140*x^13 - 2184*x^11 + 24310*x^9 - 177320*x^7 + 754572*x^5 - 1529080*x^3 + 929569*x
17:
x^17 - 17/2*x^16 + 170*x^14 - 3094*x^12 + 41327*x^10 - 376805*x^8 + 2137954*x^6 - 6498590*x^4 + 15802673/2*x^2 - 3202291/2
18:
x^18 - 9*x^17 + 204*x^15 - 4284*x^13 + 67626*x^11 - 753610*x^9 + 5497596*x^7 - 23394924*x^5 + 47408019*x^3 - 28820619*x
19:
x^19 - 19/2*x^18 + 969/4*x^16 - 5814*x^14 + 214149/2*x^12 - 1431859*x^10 + 26113581/2*x^8 - 74083926*x^6 + 900752361/4*x^4 - 547591761/2*x^2 + 221930581/4
20:
x^20 - 10*x^19 + 285*x^17 - 7752*x^15 + 164730*x^13 - 2603380*x^11 + 29015090*x^9 - 211668360*x^7 + 900752361*x^5 - 1825305870*x^3 + 1109652905*x
21:
x^21 - 21/2*x^20 + 665/2*x^18 - 20349/2*x^16 + 247095*x^14 - 4555915*x^12 + 60931689*x^10 - 555629445*x^8 + 6305266527/2*x^6 - 19165711635/2*x^4 + 23302711005/2*x^2 - 4722116521/2
22:
x^22 - 11*x^21 + 385*x^19 - 13167*x^17 + 362406*x^15 - 7710010*x^13 + 121863378*x^11 - 1358205310*x^9 + 9908275971*x^7 - 42164565597*x^5 + 85443273685*x^3 - 51943281731*x
23:
x^23 - 23/2*x^22 + 1771/4*x^20 - 33649/2*x^18 + 4167669/8*x^16 - 12666445*x^14 + 467142949/2*x^12 - 3123872213*x^10 + 227890347333/8*x^8 - 323261669577/2*x^6 + 1965195294755/4*x^4 - 1194695479813/2*x^2 + 968383680827/8
24:
x^24 - 12*x^23 + 506*x^21 - 21252*x^19 + 735471*x^17 - 20266312*x^15 + 431208876*x^13 - 6815721192*x^11 + 75963449111*x^9 - 554162862132*x^7 + 2358234353706*x^5 - 4778781919252*x^3 + 2905151042481*x
25:
x^25 - 25/2*x^24 + 575*x^22 - 26565*x^20 + 2042975/2*x^18 - 63332225/2*x^16 + 770015850*x^14 - 14199419150*x^12 + 379817245555/2*x^10 - 3463517888325/2*x^8 + 9825976473775*x^6 - 29867386995325*x^4 + 72628776062025/2*x^2 - 14717667114151/2
26:
x^26 - 13*x^25 + 650*x^23 - 32890*x^21 + 1397825*x^19 - 48430525*x^17 + 1334694140*x^15 - 28398838300*x^13 + 448874926565*x^11 - 5002859172025*x^9 + 36496484045450*x^7 - 155310412375690*x^5 + 314724696268775*x^3 - 191329672483963*x
27:
x^27 - 27/2*x^26 + 2925/4*x^24 - 40365*x^22 + 7548255/4*x^20 - 145291575/2*x^18 + 9009185445/4*x^16 - 54769188150*x^14 + 4039874339085/4*x^12 - 27015439528935/2*x^10 + 492702534613575/4*x^8 - 698896855690605*x^6 + 8497566799256925/4*x^4 - 5165901157067001/2*x^2 + 2093660879252671/4
28:
x^28 - 14*x^27 + 819*x^25 - 49140*x^23 + 2516085*x^21 - 107056950*x^19 + 3709664595*x^17 - 102235817880*x^15 + 2175316951815*x^13 - 34383286673190*x^11 + 383213082477225*x^9 - 2795587422762420*x^7 + 11896593518959695*x^5 - 24107538732979338*x^3 + 14655626154768697*x
29:
x^29 - 29/2*x^28 + 1827/2*x^26 - 118755/2*x^24 + 6633315/2*x^22 - 310465155/2*x^20 + 11953363695/2*x^18 - 370604839815/2*x^16 + 9012027371805/2*x^14 - 166185885587085/2*x^12 + 2222635878367905/2*x^10 - 20268008815027545/2*x^8 + 115000404016610385/2*x^6 - 349559311628200401/2*x^4 + 425013158488292213/2*x^2 - 86125672563201181/2
30:
x^30 - 15*x^29 + 1015*x^27 - 71253*x^25 + 4326075*x^23 - 221760825*x^21 + 9436866075*x^19 - 327004270425*x^17 + 9012027371805*x^15 - 191752944908175*x^13 + 3030867106865325*x^11 - 33780014691712575*x^9 + 246429437178450825*x^7 - 1048677934884601203*x^5 + 2125065792441461065*x^3 - 1291885088448017715*x
31:
x^31 - 31/2*x^30 + 4495/4*x^28 - 169911/2*x^26 + 44702775/8*x^24 - 624962325/2*x^22 + 58508569665/4*x^20 - 1126348042575/2*x^18 + 279372848525955/16*x^16 - 849191613164775/2*x^14 + 31318960104275025/4*x^12 - 209436091088617965/2*x^10 + 7639312552531975575/8*x^8 - 10836338660474212431/2*x^6 + 65877039565685293015/4*x^4 - 40048437741888549165/2*x^2 + 129848163681107301953/32
32:
x^32 - 16*x^31 + 1240*x^29 - 100688*x^27 + 7152444*x^25 - 434756400*x^23 + 22288978920*x^21 - 948503614800*x^19 + 32867393944230*x^17 - 905804387375760*x^15 + 19273206218015400*x^13 - 304634314310717040*x^11 + 3395250023347544700*x^9 - 24768774081083914128*x^7 + 105403263305096468824*x^5 - 213591667956738928880*x^3 + 129848163681107301953*x
33:
x^33 - 33/2*x^32 + 1364*x^30 - 118668*x^28 + 9078102*x^26 - 597790050*x^24 + 33433468380*x^22 - 1565030964420*x^20 + 60256888897755*x^18 - 1868221548962505*x^16 + 45429700371036300*x^14 - 837744364354471860*x^12 + 11204325077046897510*x^10 - 102171193084471145778*x^8 + 579717948178030578532*x^6 - 1762131260643096163260*x^4 + 4284989401476540964449/2*x^2 - 868320396104950823611/2
34:
x^34 - 17*x^33 + 1496*x^31 - 139128*x^29 + 11431684*x^27 - 812994468*x^25 + 49423388040*x^23 - 2533859656680*x^21 + 107828116974930*x^19 - 3736443097925010*x^17 + 102973987507682280*x^15 - 2191023722157849480*x^13 + 34631550238144955940*x^11 - 385980062763557661828*x^9 + 2815772891150434238584*x^7 - 11982492572373053910168*x^5 + 24281606608367065465211*x^3 - 14761446733784164001387*x
35:
x^35 - 35/2*x^34 + 6545/4*x^32 - 162316*x^30 + 14289605*x^28 - 1094415630*x^26 + 72075774225*x^24 - 4031140362900*x^22 + 377398409412255/2*x^20 - 7265306023743075*x^18 + 450511195346109975/2*x^16 - 5477559305394623700*x^14 + 101008688194589454825*x^12 - 1350930219672451816398*x^10 + 12319006398783149793805*x^8 - 69897873338842814475980*x^6 + 849856231292847291282385/4*x^4 - 516650635682445740048545/2*x^2 + 209390615747646519456961/4
36:
x^36 - 18*x^35 + 1785*x^33 - 188496*x^31 + 17738820*x^29 - 1459220840*x^27 + 103789114884*x^25 - 6309611002800*x^23 + 323484350924790*x^21 - 13765842992355300*x^19 + 477011853895881150*x^17 - 13146142332947096880*x^15 + 279716367308093874900*x^13 - 4421226173473478671848*x^11 + 49276025595132599175220*x^9 - 359474777171191617305040*x^7 + 1529741216327125124308293*x^5 - 3099903814094674440291270*x^3 + 1884515541728818675112649*x
37:
x^37 - 37/2*x^36 + 3885/2*x^34 - 435897/2*x^32 + 21877878*x^30 - 1928256110*x^28 + 147699894258*x^26 - 9727316962650*x^24 + 544041862918965*x^22 - 25466809535857305*x^20 + 980524366341533475*x^18 - 30400454144940161535*x^16 + 739250399314248097950*x^14 - 13632114034876559238198*x^12 + 182321294701990616948314*x^10 - 1662570844416761230035810*x^8 + 18866808334701209866468947/2*x^6 - 57348220560751477145388495/2*x^4 + 69727075043966290979168013/2*x^2 - 14129659550745551130667441/2
38:
x^38 - 19*x^37 + 2109*x^35 - 250971*x^33 + 26818044*x^31 - 2526680420*x^29 + 207873925252*x^27 - 14785521783228*x^25 + 898851773518290*x^23 - 46082798207741790*x^21 + 1961048732683066950*x^19 - 67953956323983890490*x^17 + 1872767678262761848140*x^15 - 39847717948100711619348*x^13 + 629837199879603949457812*x^11 - 7019743565315214082373420*x^9 + 51209908337046141066129999*x^7 - 217923238130855613152476281*x^5 + 441604808611786509534730749*x^3 - 268463531464165471482681379*x
39:
x^39 - 39/2*x^38 + 9139/4*x^36 - 575757/2*x^34 + 261475929/8*x^32 - 3284684546*x^30 + 289538681601*x^28 - 22178282674842*x^26 + 5842536527868885/4*x^24 - 81692233186451355*x^22 + 7648090057463961105/2*x^20 - 147233572035298429395*x^18 + 18259484863061928019365/4*x^16 - 111004357141137696653898*x^14 + 2046970899608712835737889*x^12 - 27376999904729334921256338*x^10 + 1997186425144799501579069961/8*x^8 - 2833002095701122970982191653/2*x^6 + 17222587535859673871854499211/4*x^4 - 10470077727102453387824573781/2*x^2 + 8486725345098385062639014237/8
40:
x^40 - 20*x^39 + 2470*x^37 - 329004*x^35 + 39617565*x^33 - 4238302640*x^31 + 399363698760*x^29 - 32856715073840*x^27 + 2337014611147554*x^25 - 142073449019915400*x^23 + 7283895292822820100*x^21 - 309965414811154588200*x^19 + 10740873448859957658450*x^17 - 296011619043033857743728*x^15 + 6298371998796039494578120*x^13 - 99552726926288490622750320*x^11 + 1109548013969333056432816645*x^9 - 8094291702003208488520547580*x^7 + 34445175071719347743708998422*x^5 - 69800518180683022585497158540*x^3 + 42433626725491925313195071185*x
41:
x^41 - 41/2*x^40 + 2665*x^38 - 374699*x^36 + 95548245/2*x^34 - 10860650515/2*x^32 + 545797054972*x^30 - 48111618500980*x^28 + 3685292271424989*x^26 - 242708808742355475*x^24 + 13574532136624346550*x^22 - 635429100362866905810*x^20 + 24465322855736570222025*x^18 - 758529773797774260468303*x^16 + 18445232282188401376978780*x^14 - 340138483664819009627730260*x^12 + 9098293714548531062749096489/2*x^10 - 82966489945532887007335612695/2*x^8 + 235375362990082209582011489217*x^6 - 715455311352000981501345875035*x^4 + 1739778695745168937840997918585/2*x^2 - 352552873457246307069012458671/2
42:
x^42 - 21*x^41 + 2870*x^39 - 425334*x^37 + 57328947*x^35 - 6911323055*x^33 + 739466977704*x^31 - 69678895760040*x^29 + 5732676866661094*x^27 - 407750798687157198*x^25 + 24788276075574893700*x^23 - 1270858200725733811620*x^21 + 54081239996891365753950*x^19 - 1874014735265089349392278*x^17 + 51646650390127523855540584*x^15 - 1098908947224799877258820840*x^13 + 17369469818683559301611911479*x^11 - 193588476539576736350449762955*x^9 + 1412252177940493257492068935302*x^7 - 6009824615356808244611305350294*x^5 + 12178450870216182564886985430095*x^3 - 7403610342602172448449261632091*x
43:
x^43 - 43/2*x^42 + 12341/4*x^40 - 481299*x^38 + 273904969/4*x^36 - 17481581845/2*x^34 + 3974635005159/4*x^32 - 99873083922724*x^30 + 17607507519030503/2*x^28 - 674357090136452289*x^26 + 88824655937476702425/2*x^24 - 2483950119600297904530*x^22 + 232549331986632872741985/2*x^20 - 4476812978688824556881553*x^18 + 277600745846935440723530639/2*x^16 - 3375220337904742480152092580*x^14 + 248962400734464349989770731199/4*x^12 - 1664860898240359932613867961413/2*x^10 + 30363421825720605036079482108993/4*x^8 - 43070409743390459086381021677107*x^6 + 523673387419295850290140373494085/4*x^4 - 318355244731893415283318250179913/2*x^2 + 129024520859926228378837238913451/4
44:
x^44 - 22*x^43 + 3311*x^41 - 543004*x^39 + 81431207*x^37 - 10988422874*x^35 + 1324878335053*x^33 - 141755344922576*x^31 + 13357419497195554*x^29 - 1098952295037181508*x^27 + 78165697224979498134*x^25 - 4751904576626656860840*x^23 + 243623109700282057158270*x^21 - 10367356371700435815936228*x^19 + 359248024037210570348098474*x^17 - 9900646324520577941779471568*x^15 + 210660492929162142299036772553*x^13 - 3329721796480719865227735922826*x^11 + 37110848898102961710763811466547*x^9 - 270728289815597171400109279113244*x^7 + 1152081452322450870638308821686987*x^5 - 2334605128033885045411000501319362*x^3 + 1419269729459188512167209628047961*x
45:
x^45 - 45/2*x^44 + 7095/2*x^42 - 1221759/2*x^40 + 192863385/2*x^38 - 27471057185/2*x^36 + 3507030886905/2*x^34 - 398686907594745/2*x^32 + 20036129245793331*x^30 - 1766173331309755995*x^28 + 135286783658618362155*x^26 - 8909821081174981614075*x^24 + 498319997114213298732825*x^22 - 23326551836325980585856513*x^20 + 898120060093026425870246185*x^18 - 27845567787714125461254763785*x^16 + 1354246025973185200493807823555/2*x^14 - 24972913473605398989208019421195/2*x^12 + 333997640082926655396874303198923/2*x^10 - 3045693260425468178251229390023995/2*x^8 + 17281221784836763059574632325304805/2*x^6 - 52528615380762413521747511279685645/2*x^4 + 63867137825663483047524433262158245/2*x^2 - 12942188000689093683411117827763301/2
46:
x^46 - 23*x^45 + 3795*x^43 - 685377*x^41 + 113739945*x^39 - 17076603115*x^37 + 2304620297109*x^35 - 277872693172095*x^33 + 29731030493757846*x^31 - 2801516318629268130*x^29 + 230488594381349802190*x^27 - 16394070789361966169898*x^25 + 996639994228426597465650*x^23 - 51096256403380719378542838*x^21 + 2174395934962063978422701290*x^19 - 75346830484402927718689360830*x^17 + 2076510573158883974090505329451*x^15 - 44182846914840321288598803591345*x^13 + 698358701991573915829828088506839*x^11 - 7783438332198418677753141774505765*x^9 + 56781157293035078624316649068858645*x^7 - 241631630751507102200038551886553967*x^5 + 489648056663420036697687321676546545*x^3 - 297670324015849154718455710038555923*x
47:
x^47 - 47/2*x^46 + 16215/4*x^44 - 1533939/2*x^42 + 1069155483/8*x^40 - 42242123495/2*x^38 + 12035239329347/4*x^36 - 768236269358145/2*x^34 + 698679216603309381/16*x^32 - 4389042232519186737*x^30 + 773783138280245764495/2*x^28 - 29635435657692784999431*x^26 + 7807013288122675013480925/4*x^24 - 109160184134495173217796063*x^22 + 10219660894321700698586696063/2*x^20 - 196738946264829866821022219945*x^18 + 97595996938467546782253750484197/16*x^16 - 296656257856785014366306252684745/2*x^14 + 10940952997867991348000640053273811/4*x^12 - 73164320322665135570879532680354191/2*x^10 + 2668714392772648695342882506236356315/8*x^8 - 3785562215106944601133937312889345483/2*x^6 + 23013458663180741724791304118797687615/4*x^4 - 13990505228744910271767418371812128381/2*x^2 + 22680552792491997823522126468923904459/16
48:
x^48 - 24*x^47 + 4324*x^45 - 856152*x^43 + 156461778*x^41 - 25995152920*x^39 + 3903320863572*x^37 - 526790584702728*x^35 + 63516292418482671*x^33 - 6795936360029708496*x^31 + 640372252369858563720*x^29 - 52685218947009395554544*x^27 + 3747366378298884006470844*x^25 - 227812558193729057150183088*x^23 + 11679612450653372226956224072*x^21 - 497024706353254400389950871440*x^19 + 17222822989141331785103603026623*x^17 - 474650012570856022986090004295592*x^15 + 10099341228801222782769821587637364*x^13 - 159631244340360295791009889484409144*x^11 + 1779142928515099130228588337490904210*x^9 - 12979070451795238632459213644192041656*x^7 + 55232300791633780139499129885114450276*x^5 - 111924041829959282174139346974497027048*x^3 + 68041658377475993470566379406771713377*x
49:
x^49 - 49/2*x^48 + 4606*x^46 - 953442*x^44 + 182538741*x^42 - 31844062327*x^40 + 5033229534606*x^38 - 717020518067602*x^36 + 183076372265038287/2*x^34 - 20812555102590982269/2*x^32 + 1045941345537435654076*x^30 - 92199133157266442220452*x^28 + 7062344328332512166041206*x^26 - 465117306312196825014957138*x^24 + 26013682276455238141857044524*x^22 - 1217710530565473280955379635028*x^20 + 93768702940880584163341838700503/2*x^18 - 2907231326996493140789801276310501/2*x^16 + 35347694300804279739694375556730774*x^14 - 651827581056471207813290382061337338*x^12 + 8717800349723985738120082853705430629*x^10 - 79496806517245836623812683570676255143*x^8 + 451063789798342537805909560728434677254*x^6 - 1371069512417001206633207000437588581338*x^4 + 3334041260496323680057752590931813955473/2*x^2 - 675618013651758631167025175564066787331/2
50:
x^50 - 25*x^49 + 4900*x^47 - 1059380*x^45 + 212254350*x^43 - 38834222350*x^41 + 6452858377700*x^39 - 968946646037300*x^37 + 130768837332170205*x^35 - 15767087198932562325*x^33 + 1687002170221670409800*x^31 - 158964022684942141759400*x^29 + 13078415422837985492668900*x^27 - 930234612624393650029914276*x^25 + 56551483209685300308384879400*x^23 - 2899310787060650668941380083400*x^21 + 123379872290632347583344524605925*x^19 - 4275340186759548736455590112221325*x^17 + 117825647669347599132314585189102580*x^15 - 2507029157909504645435732238697451300*x^13 + 39626365226018116991454922062297411950*x^11 - 441648925095810203465626019837090306350*x^9 + 3221884212845303841470782576631676266100*x^7 - 13710695124170012066332070004375885813380*x^5 + 27783677170802697333814604924431782962275*x^3 - 16890450341293965779175629389101669683275*x
Definition
The Euler polynomials are defined by the exponential generating function $\frac{2 e^{xt}}{e^t + 1} = \sum_{n=0}^{\infty} E_n(x) \frac{t^n}{n!}$. Equivalently $E_0 = 1$ and, for $n \geq 1$, $E_n$ is the polynomial of degree $n$ with $E_n'(x) = n E_{n-1}(x)$ and $E_n(x) + E_n(x+1) = 2x^n$.
Parameters
$n$
—   integer ($0 \leq n \leq 50$)
Formulas
(1)
$E_n(x) + E_n(x+1) = 2x^{n}$.
(2)
$E_n'(x) = n E_{n-1}(x)$.
(3)
$E_n(1-x) = (-1)^n E_n(x)$.
(4)
$E_n = 2^{n} E_n\!\left(\tfrac{1}{2}\right)$ are the Euler numbers, which are integers and vanish for odd $n$.
(5)
$E_n(x) = \frac{2}{n+1}\left(B_{n+1}(x) - 2^{n+1} B_{n+1}\!\left(\tfrac{x}{2}\right)\right)$, where $B_n$ are the Bernoulli polynomials Bernoulli_polynomials.
(6)
$\sum_{k=0}^{m-1} (-1)^{k} k^{n} = \frac{E_n(0) + (-1)^{m-1} E_n(m)}{2}$, the alternating counterpart of Faulhaber's formula.
Comments
(7)
The Bernoulli polynomials Bernoulli_polynomials are the companion family: the generating function has $e^t - 1$ in place of $e^t + 1$, and they sum powers where these sum them alternately.
(8)
$E_n(0)$ is rational and $E_n(\tfrac{1}{2})$ is a rational whose numerator is the Euler number; the polynomials themselves have rational coefficients with denominators a power of two.
Programs
(P1)
Sage
R. = QQ[]
def euler_polynomial(n):
    B = lambda m, y: bernoulli_polynomial(y, m)
    return R(2/(n+1) * (B(n+1, x) - 2^(n+1) * B(n+1, x/2)))

euler_polynomial(51)             # the next one after this table
Links
Data properties
Entries are of type: rational polynomial
Table is complete: false