Sums of powers $S_p(n)$

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Polynomials

$p$

$S_p(n)$

1/2*n^2 + 1/2*n

1/3*n^3 + 1/2*n^2 + 1/6*n

1/4*n^4 + 1/2*n^3 + 1/4*n^2

1/5*n^5 + 1/2*n^4 + 1/3*n^3 - 1/30*n

1/6*n^6 + 1/2*n^5 + 5/12*n^4 - 1/12*n^2

1/7*n^7 + 1/2*n^6 + 1/2*n^5 - 1/6*n^3 + 1/42*n

1/8*n^8 + 1/2*n^7 + 7/12*n^6 - 7/24*n^4 + 1/12*n^2

1/9*n^9 + 1/2*n^8 + 2/3*n^7 - 7/15*n^5 + 2/9*n^3 - 1/30*n

1/10*n^10 + 1/2*n^9 + 3/4*n^8 - 7/10*n^6 + 1/2*n^4 - 3/20*n^2

10:

1/11*n^11 + 1/2*n^10 + 5/6*n^9 - n^7 + n^5 - 1/2*n^3 + 5/66*n

11:

1/12*n^12 + 1/2*n^11 + 11/12*n^10 - 11/8*n^8 + 11/6*n^6 - 11/8*n^4 + 5/12*n^2

12:

1/13*n^13 + 1/2*n^12 + n^11 - 11/6*n^9 + 22/7*n^7 - 33/10*n^5 + 5/3*n^3 - 691/2730*n

13:

1/14*n^14 + 1/2*n^13 + 13/12*n^12 - 143/60*n^10 + 143/28*n^8 - 143/20*n^6 + 65/12*n^4 - 691/420*n^2

14:

1/15*n^15 + 1/2*n^14 + 7/6*n^13 - 91/30*n^11 + 143/18*n^9 - 143/10*n^7 + 91/6*n^5 - 691/90*n^3 + 7/6*n

15:

1/16*n^16 + 1/2*n^15 + 5/4*n^14 - 91/24*n^12 + 143/12*n^10 - 429/16*n^8 + 455/12*n^6 - 691/24*n^4 + 35/4*n^2

16:

1/17*n^17 + 1/2*n^16 + 4/3*n^15 - 14/3*n^13 + 52/3*n^11 - 143/3*n^9 + 260/3*n^7 - 1382/15*n^5 + 140/3*n^3 - 3617/510*n

17:

1/18*n^18 + 1/2*n^17 + 17/12*n^16 - 17/3*n^14 + 221/9*n^12 - 2431/30*n^10 + 1105/6*n^8 - 11747/45*n^6 + 595/3*n^4 - 3617/60*n^2

18:

1/19*n^19 + 1/2*n^18 + 3/2*n^17 - 34/5*n^15 + 34*n^13 - 663/5*n^11 + 1105/3*n^9 - 23494/35*n^7 + 714*n^5 - 3617/10*n^3 + 43867/798*n

19:

1/20*n^20 + 1/2*n^19 + 19/12*n^18 - 323/40*n^16 + 323/7*n^14 - 4199/20*n^12 + 4199/6*n^10 - 223193/140*n^8 + 2261*n^6 - 68723/40*n^4 + 43867/84*n^2

20:

1/21*n^21 + 1/2*n^20 + 5/3*n^19 - 19/2*n^17 + 1292/21*n^15 - 323*n^13 + 41990/33*n^11 - 223193/63*n^9 + 6460*n^7 - 68723/10*n^5 + 219335/63*n^3 - 174611/330*n

21:

1/22*n^22 + 1/2*n^21 + 7/4*n^20 - 133/12*n^18 + 323/4*n^16 - 969/2*n^14 + 146965/66*n^12 - 223193/30*n^10 + 33915/2*n^8 - 481061/20*n^6 + 219335/12*n^4 - 1222277/220*n^2

22:

1/23*n^23 + 1/2*n^22 + 11/6*n^21 - 77/6*n^19 + 209/2*n^17 - 3553/5*n^15 + 11305/3*n^13 - 223193/15*n^11 + 124355/3*n^9 - 755953/10*n^7 + 482537/6*n^5 - 1222277/30*n^3 + 854513/138*n

23:

1/24*n^24 + 1/2*n^23 + 23/12*n^22 - 1771/120*n^20 + 4807/36*n^18 - 81719/80*n^16 + 37145/6*n^14 - 5133439/180*n^12 + 572033/6*n^10 - 17386919/80*n^8 + 11098351/36*n^6 - 28112371/120*n^4 + 854513/12*n^2

24:

1/25*n^25 + 1/2*n^24 + 2*n^23 - 253/15*n^21 + 506/3*n^19 - 14421/10*n^17 + 29716/3*n^15 - 10266878/195*n^13 + 208012*n^11 - 17386919/30*n^9 + 22196702/21*n^7 - 28112371/25*n^5 + 1709026/3*n^3 - 236364091/2730*n

25:

1/26*n^26 + 1/2*n^25 + 25/12*n^24 - 115/6*n^22 + 1265/6*n^20 - 24035/12*n^18 + 185725/12*n^16 - 25667195/273*n^14 + 1300075/3*n^12 - 17386919/12*n^10 + 277458775/84*n^8 - 28112371/6*n^6 + 21362825/6*n^4 - 1181820455/1092*n^2

26:

1/27*n^27 + 1/2*n^26 + 13/6*n^25 - 65/3*n^23 + 16445/63*n^21 - 16445/6*n^19 + 142025/6*n^17 - 10266878/63*n^15 + 2600150/3*n^13 - 20548177/6*n^11 + 3606964075/378*n^9 - 52208689/3*n^7 + 55543345/3*n^5 - 1181820455/126*n^3 + 8553103/6*n

27:

1/28*n^28 + 1/2*n^27 + 9/4*n^26 - 195/8*n^24 + 4485/14*n^22 - 29601/8*n^20 + 142025/4*n^18 - 15400317/56*n^16 + 1671525*n^14 - 61644531/8*n^12 + 721392815/28*n^10 - 469878201/8*n^8 + 166630035/2*n^6 - 3545461365/56*n^4 + 76977927/4*n^2

28:

1/29*n^29 + 1/2*n^28 + 7/3*n^27 - 273/10*n^25 + 390*n^23 - 9867/2*n^21 + 52325*n^19 - 905901/2*n^17 + 3120180*n^15 - 33193209/2*n^13 + 65581165*n^11 - 365460823/2*n^9 + 333260070*n^7 - 709092273/2*n^5 + 179615163*n^3 - 23749461029/870*n

29:

1/30*n^30 + 1/2*n^29 + 29/12*n^28 - 609/20*n^26 + 1885/4*n^24 - 26013/4*n^22 + 303485/4*n^20 - 8757043/12*n^18 + 22621305/4*n^16 - 137514723/4*n^14 + 1901853785/12*n^12 - 10598363867/20*n^10 + 4832271015/4*n^8 - 6854558639/4*n^6 + 5208839727/4*n^4 - 23749461029/60*n^2

30:

1/31*n^31 + 1/2*n^30 + 5/2*n^29 - 203/6*n^27 + 1131/2*n^25 - 16965/2*n^23 + 216775/2*n^21 - 2304485/2*n^19 + 19959975/2*n^17 - 137514723/2*n^15 + 731482225/2*n^13 - 31795091601/22*n^11 + 8053785025/2*n^9 - 102818379585/14*n^7 + 15626519181/2*n^5 - 23749461029/6*n^3 + 8615841276005/14322*n

31:

1/32*n^32 + 1/2*n^31 + 31/12*n^30 - 899/24*n^28 + 2697/4*n^26 - 175305/16*n^24 + 6720025/44*n^22 - 14287807/8*n^20 + 68751025/4*n^18 - 4262956413/32*n^16 + 22675948975/28*n^14 - 328549279877/88*n^12 + 49933467155/4*n^10 - 3187369767135/112*n^8 + 161474031537/4*n^6 - 736233291899/24*n^4 + 8615841276005/924*n^2

32:

1/33*n^33 + 1/2*n^32 + 8/3*n^31 - 124/3*n^29 + 7192/9*n^27 - 70122/5*n^25 + 2337400/11*n^23 - 57151228/21*n^21 + 28947800*n^19 - 4262956413/17*n^17 + 36281518360/21*n^15 - 101092086116/11*n^13 + 36315248840*n^11 - 2124913178090/21*n^9 + 184541750328*n^7 - 2944933167596/15*n^5 + 68926730208040/693*n^3 - 7709321041217/510*n

33:

1/34*n^34 + 1/2*n^33 + 11/4*n^32 - 682/15*n^30 + 19778/21*n^28 - 89001/5*n^26 + 292175*n^24 - 28575614/7*n^22 + 47763870*n^20 - 15630840181/34*n^18 + 49887087745/14*n^16 - 21662589882*n^14 + 99866934310*n^12 - 2337404495899/7*n^10 + 761234720103*n^8 - 16197132421778/15*n^6 + 17231682552010/21*n^4 - 84802531453387/340*n^2

34:

1/35*n^35 + 1/2*n^34 + 17/6*n^33 - 748/15*n^31 + 23188/21*n^29 - 336226/15*n^27 + 397358*n^25 - 42242212/7*n^23 + 77331980*n^21 - 822675799*n^19 + 49887087745/7*n^17 - 245509351996/5*n^15 + 261190443580*n^13 - 7224704805506/7*n^11 + 2875775609278*n^9 - 78671786048636/15*n^7 + 117175441353668/21*n^5 - 84802531453387/30*n^3 + 2577687858367/6*n

35:

1/36*n^36 + 1/2*n^35 + 35/12*n^34 - 1309/24*n^32 + 11594/9*n^30 - 168113/6*n^28 + 534905*n^26 - 52802765/6*n^24 + 123028150*n^22 - 5758730593/4*n^20 + 249435438725/18*n^18 - 429641365993/4*n^16 + 652976108950*n^14 - 18061762013765/6*n^12 + 10065214632473*n^10 - 137675625585113/6*n^8 + 292938603384170/9*n^6 - 593617720173709/24*n^4 + 90219075042845/12*n^2

36:

1/37*n^37 + 1/2*n^36 + 3*n^35 - 119/2*n^33 + 1496*n^31 - 34782*n^29 + 2139620/3*n^27 - 63363318/5*n^25 + 192565800*n^23 - 2468027397*n^21 + 498870877450/19*n^19 - 227457193761*n^17 + 1567142661480*n^15 - 108370572082590/13*n^13 + 32940702433548*n^11 - 275351251170226/3*n^9 + 1171754413536680/7*n^7 - 1780853160521127/10*n^5 + 90219075042845*n^3 - 26315271553053477373/1919190*n

37:

1/38*n^38 + 1/2*n^37 + 37/12*n^36 - 259/4*n^34 + 6919/4*n^32 - 214489/5*n^30 + 2827355/3*n^28 - 1172221383/65*n^26 + 296872275*n^24 - 8301546699/2*n^22 + 1845822246565/38*n^20 - 935101796573/2*n^18 + 7248034809345/2*n^16 - 2004855583527915/91*n^14 + 101567165836773*n^12 - 5093998146649181/15*n^10 + 5419364162607145/7*n^8 - 21963855646427233/20*n^6 + 3338105776585265/4*n^4 - 26315271553053477373/103740*n^2

38:

1/39*n^39 + 1/2*n^38 + 19/6*n^37 - 703/10*n^35 + 11951/6*n^33 - 262922/5*n^31 + 3704810/3*n^29 - 14848137518/585*n^27 + 451245858*n^25 - 6857799447*n^23 + 1845822246565/21*n^21 - 935101796573*n^19 + 8100744786915*n^17 - 5078967478270718/91*n^15 + 296888638599798*n^13 - 17597448142969898/15*n^11 + 205935838179071510/63*n^9 - 59616179611731061/10*n^7 + 12684801951024007/2*n^5 - 26315271553053477373/8190*n^3 + 2929993913841559/6*n

39:

1/40*n^40 + 1/2*n^39 + 13/4*n^38 - 9139/120*n^36 + 9139/4*n^34 - 5126979/80*n^32 + 4816253/3*n^30 - 7424068759/210*n^28 + 676868787*n^26 - 89151392811/8*n^24 + 2181426291395/14*n^22 - 36468970066347/20*n^20 + 35103227409965/2*n^18 - 7618451217406077/56*n^16 + 827046921813723*n^14 - 114383412929304337/30*n^12 + 267716589632792963/21*n^10 - 2325031004857511379/80*n^8 + 164902425363312091/4*n^6 - 26315271553053477373/840*n^4 + 38089920879940267/4*n^2

40:

1/41*n^41 + 1/2*n^40 + 10/3*n^39 - 247/3*n^37 + 18278/7*n^35 - 155363/2*n^33 + 6214520/3*n^31 - 1024009484/21*n^29 + 3008305720/3*n^27 - 89151392811/5*n^25 + 1896892427300/7*n^23 - 3473235244414*n^21 + 36950765694700*n^19 - 2240720946295905/7*n^17 + 2205458458169928*n^15 - 35194896285939796/3*n^13 + 10708663585311718520/231*n^11 - 775010334952503793/6*n^9 + 235574893376160130*n^7 - 26315271553053477373/105*n^5 + 380899208799402670/3*n^3 - 261082718496449122051/13530*n

41:

1/42*n^42 + 1/2*n^41 + 41/12*n^40 - 533/6*n^38 + 374699/126*n^36 - 374699/4*n^34 + 31849415/12*n^32 - 20992194422/315*n^30 + 4405019090/3*n^28 - 281169777327/10*n^26 + 19443147379825/42*n^24 - 71201322510487/11*n^22 + 75749069674135*n^20 - 30623186266044035/42*n^18 + 11302974598120881/2*n^16 - 103070767694537974/3*n^14 + 109763801749445114830/693*n^12 - 31775423733052655513/60*n^10 + 4829285314211282665/4*n^8 - 1078926133675192572293/630*n^6 + 7808433780387754735/6*n^4 - 261082718496449122051/660*n^2

42:

1/43*n^43 + 1/2*n^42 + 7/2*n^41 - 287/3*n^39 + 10127/3*n^37 - 1124097/10*n^35 + 222945905/66*n^33 - 1354335124/15*n^31 + 2126560940*n^29 - 656062813763/15*n^27 + 777725895193*n^25 - 130019806323498/11*n^23 + 151498139348270*n^21 - 1611746645581265*n^19 + 13962498032972853*n^17 - 1442990747723531636/15*n^15 + 16886738730683863820/33*n^13 - 20220724193760780781/10*n^11 + 33804997199478978655/6*n^9 - 1078926133675192572293/105*n^7 + 10931807292542856629*n^5 - 1827579029475143854357/330*n^3 + 1520097643918070802691/1806*n

43:

1/44*n^44 + 1/2*n^43 + 43/12*n^42 - 12341/120*n^40 + 22919/6*n^38 - 16112057/120*n^36 + 563921995/132*n^34 - 14559102583/120*n^32 + 9144212042/3*n^30 - 4030100141687/60*n^28 + 2572477961023/2*n^26 - 931808611985069/44*n^24 + 296109999635255*n^22 - 13861021151998879/4*n^20 + 66709712824203631/2*n^18 - 15512150538027965087/60*n^16 + 363064882709703072130/231*n^14 - 869491140331713573583/120*n^12 + 290722975915519216433/12*n^10 - 46393823748033280608599/840*n^8 + 470067713579342835047/6*n^6 - 78585898267431185737351/1320*n^4 + 1520097643918070802691/84*n^2

44:

1/45*n^45 + 1/2*n^44 + 11/3*n^43 - 3311/30*n^41 + 38786/9*n^39 - 4790071/30*n^37 + 16112057/3*n^35 - 14559102583/90*n^33 + 12978881608/3*n^31 - 1528658674433/15*n^29 + 56594515142506/27*n^27 - 931808611985069/25*n^25 + 13028839983951220/23*n^23 - 152471232671987669/21*n^21 + 77242825375393678*n^19 - 10037273877547506821/15*n^17 + 290451906167762457704/63*n^15 - 735723272588373023801/30*n^13 + 290722975915519216433/3*n^11 - 510332061228366086694589/1890*n^9 + 1477355671249363195862/3*n^7 - 78585898267431185737351/150*n^5 + 16721074083098778829601/63*n^3 - 27833269579301024235023/690*n

45:

1/46*n^46 + 1/2*n^45 + 15/4*n^44 - 473/4*n^42 + 19393/4*n^40 - 756327/4*n^38 + 80560285/12*n^36 - 856417799/4*n^34 + 24335403015/4*n^32 - 1528658674433/10*n^30 + 141486287856265/42*n^28 - 645098269835817/10*n^26 + 48858149939817075/46*n^24 - 207915317279983185/14*n^22 + 347592714189271551/2*n^20 - 10037273877547506821/6*n^18 + 181532441354851536065/14*n^16 - 315309973966445581629/4*n^14 + 1453614879577596082165/4*n^12 - 510332061228366086694589/420*n^10 + 11080167534370223968965/4*n^8 - 78585898267431185737351/20*n^6 + 83605370415493894148005/28*n^4 - 83499808737903072705069/92*n^2

46:

1/47*n^47 + 1/2*n^46 + 23/6*n^45 - 253/2*n^43 + 10879/2*n^41 - 446039/2*n^39 + 50078015/6*n^37 - 19697609377/70*n^35 + 16961038465/2*n^33 - 1134166113289/5*n^31 + 112213262782555/21*n^29 - 1648584467358199/15*n^27 + 1954325997592683*n^25 - 207915317279983185/7*n^23 + 380696782207297413*n^21 - 12150384167557508257/3*n^19 + 245602714774210901735/7*n^17 - 2417376467076082792489/10*n^15 + 2571780171560362299215/2*n^13 - 1067057946204765453997777/210*n^11 + 84947951096838383762065/6*n^9 - 258210808592988181708439/10*n^7 + 384584703911271913080823/14*n^5 - 27833269579301024235023/2*n^3 + 596451111593912163277961/282*n

47:

1/48*n^48 + 1/2*n^47 + 47/12*n^46 - 1081/8*n^44 + 511313/84*n^42 - 20963833/80*n^40 + 123877195/12*n^38 - 925787640719/2520*n^36 + 46892282815/4*n^34 - 53305807324583/160*n^32 + 1054804670156017/126*n^30 - 11069067137976479/60*n^28 + 7065640145142777/2*n^26 - 3257339970719736565/56*n^24 + 1626613523976634401/2*n^22 - 571068055875202888079/60*n^20 + 11543327594387912381545/126*n^18 - 113616693952575891246983/160*n^16 + 17267666866191004009015/4*n^14 - 50151723471623976337895519/2520*n^12 + 798510740310280807363411/12*n^10 - 12135908003870444540296633/80*n^8 + 18075481083829779914798681/84*n^6 - 1308163670227148139046081/8*n^4 + 596451111593912163277961/12*n^2

48:

1/49*n^49 + 1/2*n^48 + 4*n^47 - 2162/15*n^45 + 47564/7*n^43 - 1533939/5*n^41 + 38116060/3*n^39 - 50042575174/105*n^37 + 16077354108*n^35 - 4845982484053/10*n^33 + 272207656814456/21*n^31 - 1526767881100204/5*n^29 + 18841707053714072/3*n^27 - 3908807964863683878/35*n^25 + 1697335851106053288*n^23 - 326324603357258793188/15*n^21 + 4860348460794910476440/21*n^19 - 340850081857727673740949/170*n^17 + 13814133492952803207212*n^15 - 100303446943247952675791038/1365*n^13 + 290367541931011202677604*n^11 - 12135908003870444540296633/15*n^9 + 72301924335319119659194724/49*n^7 - 7848982021362888834276486/5*n^5 + 2385804446375648653111844/3*n^3 - 5609403368997817686249127547/46410*n

49:

1/50*n^50 + 1/2*n^49 + 49/12*n^48 - 2303/15*n^46 + 7567*n^44 - 3579191/10*n^42 + 93384347/6*n^40 - 9218369111/15*n^38 + 65649195941/3*n^36 - 237453141718597/340*n^34 + 238181699712649/12*n^32 - 37405813086954998/75*n^30 + 32972987343999626/3*n^28 - 13680827877022893573/65*n^26 + 3465394029341525463*n^24 - 726813889295712766646/15*n^22 + 1701121961278218666754/3*n^20 - 5567218003676218671102167/1020*n^18 + 169223135288671839288347/4*n^16 - 50151723471623976337895519/195*n^14 + 3557002388654887232800649/3*n^12 - 594659492189651782474535017/150*n^10 + 18075481083829779914798681/2*n^8 - 64100019841130258813257969/5*n^6 + 29226104468101696000620089/3*n^4 - 39265823582984723803743892829/13260*n^2

50:

1/51*n^51 + 1/2*n^50 + 25/6*n^49 - 490/3*n^47 + 75670/9*n^45 - 416185*n^43 + 56941675/3*n^41 - 92183691110/117*n^39 + 88715129650/3*n^37 - 33921877388371/34*n^35 + 541322044801475/18*n^33 - 2413278263674516/3*n^31 + 56849978179309700/3*n^29 - 45602759590076311910/117*n^27 + 6930788058683050926*n^25 - 316006038824222942020/3*n^23 + 12150871151987276191100/9*n^21 - 1465057369388478597658465/102*n^19 + 248857551895105646012275/2*n^17 - 100303446943247952675791038/117*n^15 + 13680778417903412433848650/3*n^13 - 594659492189651782474535017/33*n^11 + 451887027095744497869967025/9*n^9 - 91571456915900369733225670*n^7 + 292261044681016960006200890/3*n^5 - 196329117914923619018719464145/3978*n^3 + 495057205241079648212477525/66*n

Definition

For an integer $p\geq 0$, $S_p(n)$ is the polynomial in $\mathbb{Q}[n]$ with $S_p(N)=\sum_{k=1}^{N} k^p$ for every positive integer $N$ [2].

Parameters

$p$

— exponent ($p\geq 0$)

Formulas

(1)

$S_p(n)=\frac{B_{p+1}(n+1)-B_{p+1}(1)}{p+1}$, where $B_m(x)$ are the Bernoulli polynomials.

(2)

$S_p(n)=\frac{1}{p+1}\sum_{j=0}^{p}(-1)^j\binom{p+1}{j}B_j n^{p+1-j}$, where $B_j$ are the Bernoulli numbers with $B_1=-\tfrac12$ [1].

(3)

$S_p(n)-S_p(n-1)=n^p$ and $S_p(0)=0$.

(4)

For $p\geq 1$, $S_p(n)/(n(n+1))\in\mathbb{Q}[n]$; for odd $p\geq 3$, $S_p(n)/(n^2(n+1)^2)\in\mathbb{Q}[n]$.

(5)

For positive integers $N$, $\sum_{p=0}^{\infty}S_p(N)\frac{t^p}{p!} =\sum_{k=1}^{N} e^{kt} =\frac{e^t(e^{Nt}-1)}{e^t-1}$.

Comments

(6)

The summation starts at $k=1$ and ends at $N$. Thus $S_0(n)=n$, not $n+1$, and the constant term of every $S_p(n)$ is zero.

(7)

The power sum symmetric polynomials share the words "power sum" but hold a different family: $p_k(x_1,\dots,x_m)=x_1^k+\cdots+x_m^k$.

(8)

For odd $p\geq 3$, $S_p(n)$ is a polynomial in the triangular number $n(n+1)/2$ and is divisible by $n^2(n+1)^2$.

Programs

(P1)

Sage

import numberdb.sage as numberdb
from sage.arith.misc import bernoulli, binomial
from sage.rings.polynomial.polynomial_ring_constructor import PolynomialRing
from sage.rings.rational_field import QQ

R = PolynomialRing(QQ, 'n')
n = R.gen()

def sums_of_powers_polynomial(p):
    value = R.zero()
    for j in range(p + 1):
        value += (QQ((-1)**j) * QQ(binomial(p + 1, j))
                  * QQ(bernoulli(j)) * n**(p + 1 - j))
    return value / QQ(p + 1)

sums_of_powers_polynomial(51)      # the next one after this table

References

[1]

R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics, second edition, Addison-Wesley, 1994, Section 6.5.

Links

[2]

Wikipedia: Faulhaber's formula

[3]

MathWorld: Faulhaber's Formula

[4]

OEIS A162298: numerators of Faulhaber's triangle for $\sum_{x=1}^{m}x^{k-1}$

[5]

OEIS A162299: denominators of Faulhaber's triangle for $\sum_{x=1}^{m}x^{k-1}$

Similar tables

Bernoulli polynomials — give $S_p(n)$ by the shifted difference in (1)

Bernoulli numbers — give the coefficients in (2)

Euler polynomials — give the corresponding polynomials for alternating sums of powers

Power sum symmetric polynomials — share the phrase "power sum" but hold symmetric polynomials in several variables

Data properties

Entries are of type: rational polynomial

How well the digits are known: exact

Table is complete: no (it holds every exponent $p$ with $0\leq p\leq 50$)

How they were obtained:

The generator uses the Bernoulli-number formula in (2), with every division made in Sage's rational field.

Before the draft was created, the entries were checked against brute-force sums for $1\leq N\leq 12$, interpolated independently from $p+2$ integer values for each $p$, and compared coefficient by coefficient with OEIS A162298 [4] and A162299 [5] through $p=50$. The finite-difference identity (3) and the divisibility statements in (4) were checked on every entry.