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Faulhaber polynomials $F_p(a)$
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Polynomials
$p$ 
$F_p(a)$
1:
a
3:
a^2
5:
4/3*a^3 - 1/3*a^2
7:
2*a^4 - 4/3*a^3 + 1/3*a^2
9:
16/5*a^5 - 4*a^4 + 12/5*a^3 - 3/5*a^2
11:
16/3*a^6 - 32/3*a^5 + 34/3*a^4 - 20/3*a^3 + 5/3*a^2
13:
64/7*a^7 - 80/3*a^6 + 656/15*a^5 - 944/21*a^4 + 2764/105*a^3 - 691/105*a^2
15:
16*a^8 - 64*a^7 + 448/3*a^6 - 704/3*a^5 + 718/3*a^4 - 140*a^3 + 35*a^2
17:
256/9*a^9 - 448/3*a^8 + 1408/3*a^7 - 9376/9*a^6 + 24304/15*a^5 - 4948/3*a^4 + 14468/15*a^3 - 3617/15*a^2
19:
256/5*a^10 - 1024/3*a^9 + 6944/5*a^8 - 144512/35*a^7 + 45264/5*a^6 - 210656/15*a^5 + 1500334/105*a^4 - 175468/21*a^3 + 43867/21*a^2
21:
1024/11*a^11 - 768*a^10 + 11776/3*a^9 - 15040*a^8 + 44096*a^7 - 3180688/33*a^6 + 24655472/165*a^5 - 5016584/33*a^4 + 4889108/55*a^3 - 1222277/55*a^2
23:
512/3*a^12 - 5120/3*a^11 + 53504/5*a^10 - 461824/9*a^9 + 968176/5*a^8 - 8485952/15*a^7 + 55602464/45*a^6 - 9576448/5*a^5 + 9742186/5*a^4 - 3418052/3*a^3 + 854513/3*a^2
25:
4096/13*a^13 - 11264/3*a^12 + 84992/3*a^11 - 166144*a^10 + 2351360/3*a^9 - 8838656/3*a^8 + 782613248/91*a^7 - 244139072/13*a^6 + 378418960/13*a^5 - 2694757500/91*a^4 + 4727281820/273*a^3 - 1181820455/273*a^2
27:
4096/7*a^14 - 8192*a^13 + 73216*a^12 - 3607552/7*a^11 + 20793344/7*a^10 - 97683456/7*a^9 + 366828480/7*a^8 - 1070551296/7*a^7 + 333946224*a^6 - 3623312928/7*a^5 + 3685987050/7*a^4 - 307911708*a^3 + 76977927*a^2
29:
16384/15*a^15 - 53248/3*a^14 + 2781184/15*a^13 - 23136256/15*a^12 + 159930368/15*a^11 - 917722624/15*a^10 + 861346816/3*a^9 - 5389667968/5*a^8 + 15728318912/5*a^7 - 34343467376/5*a^6 + 159696566288/15*a^5 - 32491767392/3*a^4 + 94997844116/15*a^3 - 23749461029/15*a^2
31:
2048*a^16 - 114688/3*a^15 + 1384448/3*a^14 - 13434880/3*a^13 + 109677568/3*a^12 - 8300750848/33*a^11 + 4325286400/3*a^10 - 6764169216*a^9 + 25393519696*a^8 - 518722631360/7*a^7 + 5339635528448/33*a^6 - 8276401041856/33*a^5 + 58936898567654/231*a^4 - 34463365104020/231*a^3 + 8615841276005/231*a^2
33:
65536/17*a^17 - 81920*a^16 + 16973824/15*a^15 - 266027008/21*a^14 + 1808441344/15*a^13 - 14691629056/15*a^12 + 706737453056/105*a^11 - 4049717698048/105*a^10 + 21531497126656/119*a^9 - 404153151102272/595*a^8 + 1179393456891008/595*a^7 - 1103677319053856/255*a^6 + 11974847812497424/1785*a^5 - 2436394987937788/357*a^4 + 339210125813548/85*a^3 - 84802531453387/85*a^2
35:
65536/9*a^18 - 524288/3*a^17 + 8216576/3*a^16 - 315293696/9*a^15 + 1153245184/3*a^14 - 3640082432*a^13 + 265814815744/9*a^12 - 608720193536/3*a^11 + 3487814311168/3*a^10 - 49086171149312/9*a^9 + 61423981402016/3*a^8 - 179246353916288/3*a^7 + 1174171924596176/9*a^6 - 606652882019296/3*a^5 + 617146060509482/3*a^4 - 360876300171380/3*a^3 + 90219075042845/3*a^2
37:
262144/19*a^19 - 1114112/3*a^18 + 6553600*a^17 - 94978048*a^16 + 5951029248/5*a^15 - 38976790528/3*a^14 + 23959044431872/195*a^13 - 14949107326976/15*a^12 + 102693350018048/15*a^11 - 11179536707696896/285*a^10 + 10489059062136320/57*a^9 - 65627358167109312/95*a^8 + 17427640274161764928/8645*a^7 - 5436265177843991152/1235*a^6 + 25278523976906619568/3705*a^5 - 36002067559622831272/5187*a^4 + 105261086212213909492/25935*a^3 - 26315271553053477373/25935*a^2
39:
131072/5*a^20 - 786432*a^19 + 232849408/15*a^18 - 1265106944/5*a^17 + 17954541568/5*a^16 - 671378931712/15*a^15 + 51232934428672/105*a^14 - 69192588099584/15*a^13 + 187065284616704/5*a^12 - 26985555344770048/105*a^11 + 154617021113754368/105*a^10 - 725336572894016512/105*a^9 + 907648842352789072/35*a^8 - 2648683750136117312/35*a^7 + 2478639708137155232/15*a^6 - 26893124699241901952/105*a^5 + 27358290741767605954/105*a^4 - 152359683519761068*a^3 + 38089920879940267*a^2
41:
1048576/21*a^21 - 4980736/3*a^20 + 36438016*a^19 - 41807052800/63*a^18 + 74115842048/7*a^17 - 3138317860864/21*a^16 + 585955226353664/315*a^15 - 141902138949632/7*a^14 + 958150419345408/5*a^13 - 489576477813769216/315*a^12 + 1761613985595296768/165*a^11 - 6423053812074235648/105*a^10 + 18079023588813304064/63*a^9 - 37705263217594711936/35*a^8 + 330092361588500219392/105*a^7 - 23785347151717985752448/3465*a^6 + 585192521545801667504/55*a^5 - 119062900437224690508/11*a^4 + 1044330873985796488204/165*a^3 - 261082718496449122051/165*a^2
43:
1048576/11*a^22 - 10485760/3*a^21 + 1272578048/15*a^20 - 1716256768*a^19 + 458776838144/15*a^18 - 80069146443776/165*a^17 + 376181684199424/55*a^16 - 4680832764936192/55*a^15 + 51006243256283136/55*a^14 - 1446466769896333312/165*a^13 + 11731474862358923776/165*a^12 - 26862576319083009024/55*a^11 + 8395209963771414016/3*a^10 - 196917221250605170688/15*a^9 + 739236160602866247296/15*a^8 - 55368774968733030174208/385*a^7 + 17271376611588141769296/55*a^6 - 26770518123001111523488/55*a^5 + 190634944188613760879062/385*a^4 - 6080390575672283210764/21*a^3 + 1520097643918070802691/21*a^2
45:
4194304/23*a^23 - 7340032*a^22 + 196083712*a^21 - 4383047680*a^20 + 86735323136*a^19 - 4610693464064/3*a^18 + 24349578100736*a^17 - 343076110581760*a^16 + 21342635906449408/5*a^15 - 976761050876317696/21*a^14 + 6595108336547958784/15*a^13 - 1230250036452516272128/345*a^12 + 59157198786475272982528/2415*a^11 - 338947867141557822731264/2415*a^10 + 106004370495715360925696/161*a^9 - 1989726014293076061704448/805*a^8 + 5806380861581619926791872/805*a^7 - 1811204792751470175718608/115*a^6 + 3930297424712502907842416/161*a^5 - 3998279145080991917093200/161*a^4 + 333999234951612290820276/23*a^3 - 83499808737903072705069/23*a^2
47:
1048576/3*a^24 - 46137344/3*a^23 + 1350565888/3*a^22 - 77464600576/7*a^21 + 3628533415936/15*a^20 - 14277754486784/3*a^19 + 26525351928332288/315*a^18 - 46677423848685568/35*a^17 + 657607896431282176/35*a^16 - 73635561250556035072/315*a^15 + 267457650495242166272/105*a^14 - 361175317399248011264/15*a^13 + 61514973343246417037312/315*a^12 - 20122303165171395129344/15*a^11 + 807051082938179749829632/105*a^10 - 3786022126776186233802752/105*a^9 + 4737632781244411587677104/35*a^8 - 2765054093555800302967744/7*a^7 + 18112773904786871659924928/21*a^6 - 4010668327615051612919744/3*a^5 + 4080040210515354431282734/3*a^4 - 2385804446375648653111844/3*a^3 + 596451111593912163277961/3*a^2
49:
16777216/25*a^25 - 96468992/3*a^24 + 15409872896/15*a^23 - 414768431104/15*a^22 + 9967730950144/15*a^21 - 72202138681344/5*a^20 + 4255216252485632/15*a^19 - 75261371792162816/15*a^18 + 6753800550957907968/85*a^17 - 3358156639787810816/3*a^16 + 1044519742049858748416/75*a^15 - 758775841106309382144/5*a^14 + 279729996670184397553664/195*a^13 - 34903512343835996041216/3*a^12 + 1198823564519313139781632/15*a^11 - 34343975858449465418466304/75*a^10 + 36519123961906588701937920/17*a^9 - 685472217237710413632776512/85*a^8 + 26004316944985566284763455104/1105*a^7 - 170343977886446361786120404576/3315*a^6 + 264032025325985183691610608688/3315*a^5 - 53719788932731298156782936612/663*a^4 + 157063294331938895214975571316/3315*a^3 - 39265823582984723803743892829/3315*a^2
Definition
For a positive odd integer $p$, the Faulhaber polynomial $F_p(a)$ is the polynomial in $\mathbb{Q}[a]$ with $F_p(N(N+1)/2)=\sum_{k=1}^{N}k^p$ for every positive integer $N$, where $a=N(N+1)/2$ is the triangular number [2].
Parameters
$p$
—   odd exponent ($p$ is a positive odd integer)
Formulas
(1)
$F_p(N(N+1)/2)=\sum_{k=1}^{N}k^p$ for positive odd $p$.
(2)
$F_p(N(N+1)/2)=S_p(N)$, where $S_p(N)$ is the sum of powers polynomial evaluated at $N$.
(3)
$F_p(N(N+1)/2)=\frac{B_{p+1}(N+1)-B_{p+1}(1)}{p+1}$, where $B_j(x)$ are the Bernoulli polynomials.
(4)
If $p=2m-1$ and $A(m,j)$ are Knuth's coefficients, then $F_p(a)=\frac{1}{2m}\sum_{j=0}^{m-1}A(m,j)(2a)^{m-j}$ [4].
(5)
For $m\geq 1$, $\sum_{k=1}^{N}k^{2m}=\frac{N+\frac12}{2m+1}F_{2m+1}'(N(N+1)/2)$, where the prime denotes differentiation with respect to $a$ [2].
(6)
$4a^3=3F_5(a)+F_3(a)$, $8a^4=4F_7(a)+4F_5(a)$, and $16a^5=5F_9(a)+10F_7(a)+F_5(a)$ [2].
(7)
$F_1(a)=a$, and for every odd $p\geq 3$ the quotient $F_p(a)/a^2$ belongs to $\mathbb{Q}[a]$.
Comments
(8)
The table is indexed by the exponent $p=2m-1$, rather than by $m$, because the exponent is what names the power sum. With $m=(p+1)/2$, the polynomial $F_p(a)$ has degree $m$.
(9)
This table uses the triangular number $a=N(N+1)/2$. Knuth writes $u=N(N+1)=2a$ [1]; the conversion from his coefficients is stated in (4).
(10)
Even power sums are not entries here, because they are not polynomials in $a$ alone. They are polynomials in $N$ and $a$, as in (5).
Programs
(P1)
Sage
import numberdb.sage as numberdb
from sage.rings.polynomial.polynomial_ring_constructor import PolynomialRing
from sage.rings.rational_field import QQ

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

def faulhaber_polynomial(p):
    degree = (p + 1) // 2
    points = []
    for N in range(degree + 1):
        x = QQ(N * (N + 1)) / QQ(2)
        y = sum(QQ(k) ** p for k in range(1, N + 1))
        points.append((x, y))
    value = R.zero()
    for i, (xi, yi) in enumerate(points):
        term = R(yi)
        for j, (xj, _) in enumerate(points):
            if i != j:
                term *= (a - QQ(xj)) / QQ(xi - xj)
        value += term
    return value

faulhaber_polynomial(51)      # the next odd exponent after this table
References
[1]
Donald E. Knuth, Johann Faulhaber and sums of powers, Mathematics of Computation 61 (1993), no. 203, 277-294. (doi)
Links
Similar tables
Sums of powers —   are the same odd sums written as polynomials in $N$, rather than as polynomials in $a=N(N+1)/2$
Bernoulli polynomials —   give the shifted-difference formula in (3)
Bernoulli numbers —   give the coefficients of the Bernoulli-polynomial formula
Data properties
Entries are of type: rational polynomial
How well the digits are known: exact
Table is complete: no (it holds every odd exponent $p$ with $1\leq p\leq 49$)
How they were obtained:

The generator constructs $F_p(a)$ by exact interpolation at $a=N(N+1)/2$ for $0\leq N\leq (p+1)/2$, with exact integer sums as the ordinates.

more

Before the draft was created, the entries were checked against (3) for every odd $p\leq 49$, against OEIS A093556 [4] and A093557 [5] for $m\leq 11$, against the small polynomials and inverse identities in [2], against (5) for $m\leq 12$, and against OEIS A000537 [6] for the cubic-sum values with $N\leq 100$.



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