A093557 -id:A093557 - cp's OEIS frontend

A196837 Coefficient table of numerator polynomials of o.g.f.s for partial sums of powers of positive integers.

1, 2, -3, 3, -12, 11, 4, -30, 70, -50, 5, -60, 255, -450, 274, 6, -105, 700, -2205, 3248, -1764, 7, -168, 1610, -7840, 20307, -26264, 13068, 8, -252, 3276, -22680, 89796, -201852, 236248, -109584, 9, -360, 6090, -56700, 316365, -1077300, 2171040, -2345400, 1026576, 10, -495, 10560, -127050, 946638, -4510275, 13667720, -25228500, 25507152, -10628640

Offset: 1

Wolfdieter Lang, Oct 10 2011

The k-th power of the positive integers has partial sums Sum_{j=1..n} j^k given as column number n >= 1, in the array A103438 (not in the triangle; see the example array given there; note that 0^0 has been set to 0 there).
The o.g.f. of column number n >= 1 of the array A103438 is obtained via Laplace transformation from the e.g.f. which is given there as
  exp(x)*(exp(n*x)-1)/(exp(x)-1) = Sum_{j=1..n} exp(j*x)
  (it is trivial that the sum is the e.g.f.).
  The o.g.f. is, therefore, Sum_{j=1..n} 1/(1-j*x), which is rewritten as P(n,x)/Product_{j=1..n} (1-j*x). This defines the row polynomials P(n,x) of the present triangle. See the link for details.
This e.g.f. - o.g.f. connection proves some conjectures by Simon Plouffe. See the o.g.f. Maple programs under, e.g., A001551(n=4) and A001552 (n=5).
This triangle organizes the sum of powers of the first n positive integers in terms of the column no. n of the Stirling2 numbers A048993 (see the formula and example given below, as well as the link).
From Wolfdieter Lang, Oct 12 2011: (Start)
With the formulas given below one finds for n >= 1, k >= 0, Sum_{j=1..n} j^k =
  Sum_{m=0..min(k,n-1)} ((n-m)*S1(n+1, n-m+1)*S2(k+n-m, n)),
  with the Stirling numbers S1 from A048994 and S2 from A048993 (this formula I did not (yet) find in the literature). See the link for the proof.
For two other formulas expressing these sums of k-th powers of the first n positive integers in terms of the row no. k of Stirling2 numbers and binomials in n see the D. E. Knuth reference given under A093556, p. 285.
  See also the given link below, eqs. (11) and (12). (End)

			n\m  0    1    2     3     4      5...
1    1
2    2   -3
3    3  -12   11
4    4  -30   70   -50
5    5  -60  255  -450   274
6    6 -105  700 -2205  3248  -1764
...
n=4 (A001551=2*A196836): the row polynomial factorizes into 2*(2-5*x)*(1-5*x+5*x^2).
n=5: 1^k + 2^k + 3^k + 4^k + 5^k, k>=0, (A001552) has as e.g.f. Sum_{j=1..5} exp(j*x). The o.g.f. is
  Sum_{j=1..5} 1/(1-j*x), and this is
  (5 - 60*x + 255*x^2 - 450*x^3 + 274*x^4)/Product_{j=1..5} (1-j*x).
n=6 (A001553): the row polynomial factorizes into
     (2 - 7*x)*(3 - 42*x + 203*x^2 - 392*x^3 + 252*x^4).
Sums of powers of the first n positive integers in terms of S2:
n=4: A001551(k) = 4*S2(k+4,4) - 30*S2(k+3,4) + 70*S2(k+2,4) - 50*S2(k+1,4), k >= 0. E.g., k=3: 4*350 - 30*65 + 70*10 - 50*1 = 100 = A001551(3).
From _Wolfdieter Lang_, Oct 12 2011: (Start)
Row polynomial for n=3: P(3,x) = (1-2*x)*(1-3*x) + (1-1*x)*(1-3*x) + (1-1*x)*(1-2*x) = 3 - 12*x + 11*x^2.
a(3,2) = +(sigma_2(2,3) + sigma_2(1,3) + sigma_2(1,2)) =
  2*3 + 1*3 + 1*2 = 11 = +1*sigma_2(1,2,3) = +1*|S1(4,4-2)|.
S1,S2 formula for sums of powers with n=4, k=3:
A001551(3) = Sum_{j=1..n} j^3 = 1*4*350 - 3*10*65 + 2*35*10 - 1*50*1 = 100. (End)

José L. Cereceda, A refinement of Lang's formula for the sum of powers of integers, arXiv:2301.02141 [math.NT], 2023.
Wolfdieter Lang, Proofs and first 15 row polynomials, 2011.

Cf. A103438, A093556/A093557 (for sums of powers).

a[n_, m_] := (n-m)*StirlingS1[n+1, n+1-m]; Flatten[ Table[ a[n, m], {n, 1, 10}, {m, 0, n-1}] ] (* Jean-François Alcover, Dec 02 2011, after Wolfdieter Lang *)

from itertools import count, islice
from sympy.functions.combinatorial.numbers import stirling
def A196837_T(n,m): return (n-m)*stirling(n+1,n+1-m,kind=1,signed=True)
def A196837_gen(): # generator of terms
    return (A196837_T(n,m) for n in count(1) for m in range(n))
A196837_list = list(islice(A196837_gen(),40)) # Chai Wah Wu, Oct 24 2024

a(n,m) = [x^m] P(n,x), m=0..n-1, with the row polynomials defined by
  (Sum_{j=1..n} 1/(1-j*x))*Product_{j=1..n} (1-j*x) (see the comment given above).
Sum_{j=1..n} j^k = Sum_{m=0..n-1} a(n,m)*S2(k+n-m,n), n >= 1, k >= 0, with the Stirling2 triangle A048993.
From Wolfdieter Lang, Oct 12 2011: (Start)
The row polynomial P(n,x) is therefore
Sum_{j=1..n} (Product_{k=1..n omitting k=j} (1-k*x)), n >= 1. This leads to:
a(n,m) = (n-m)*S1(n+1, n+1-m), n-1 >= m >= 0, with the (signed) Stirling1 numbers A048994. For the proof see the link.
(End)
A similar polynomial occurs in the expansion of 1/(n+x)^2 as a series with factorials in the denominator: 1/(n+x)^2 = -Sum_{k>=1} n!/(n+k+1)! * P(k,1/x) x^(k-1). - Matt Majic, Nov 01 2019

A093556 Triangle of numerators of coefficients of Faulhaber polynomials in Knuth's version.

Original entry on oeis.org

1, 1, 0, 1, -1, 0, 1, -4, 2, 0, 1, -5, 3, -3, 0, 1, -4, 17, -10, 5, 0, 1, -35, 287, -118, 691, -691, 0, 1, -8, 112, -352, 718, -280, 140, 0, 1, -21, 66, -293, 4557, -3711, 10851, -10851, 0, 1, -40, 217, -4516, 2829, -26332, 750167, -438670, 219335, 0, 1, -33, 506, -2585, 7579, -198793, 1540967, -627073, 1222277, -1222277, 0

Offset: 1

Wolfdieter Lang, Apr 02 2004

The companion triangle with the denominators is A093557.
In the 1986 Edwards reference, eq. 7, p. 453, the lower triangular matrix F^{-1} is obtained from F^{-1}(m,l) = A(m,m-l)/m with m >= 2, l >= 2. See the W. Lang link for this triangle.
Sum_{j=1..n} j^(2*m-1) = Sum_{k=0..m-1} A(m,k)*u^(m-k)/(2*m), with u:=n*(n+1), A(m,k):= A093556(m,k)/ A093557(m,k) and m=1,2,... (Faulhaber's m-th row polynomial in falling powers of u:=n*(n+1), divided by 2*m, gives the sum of the (2*m-1)-th power of the first n integers > 0. See the W. Lang link for the Faulhaber triangle.)
Sum_{j=1..n} j^(2*(m-1)) = (2*n+1)*Sum_{j=0..m-1} (m-j)*A(m,j)*(n*(n+1))^(m-1-j)/(2*m*(2*m-1)), with u:=n*(n+1) and m >= 2. Sum of the even powers of the first n integers > 0. From the bottom of p. 288 of the 1993 Knuth reference with A^{(m)}_k = A(m,k). See also A093558 with A093559.

			Triangle begins:
  [1];
  [1,0];
  [1,-1,0];
  [1,-4,2,0];
...
Numerators of Knuth's Faulhaber triangle A(m,k):
  [1],
  [1, 0],
  [1, -1/2, 0],
  [1, -4/3, 2/3, 0],
  ...
A(m,m-1)=1 if m=1, else 0.
Edwards' Faulhaber triangle F^{-1}(m,l) = A(m,m-l)/m, for m>=2, l>=2:
  [1/2],
  [-1/6, 1/3],
  [1/6, -1/3, 1/4],
  [-3/10, 3/5, -1/2, 1/5],
  ...

Ivo Schneider, Johannes Faulhaber 1580-1635, Birkhäuser Verlag, Basel, Boston, Berlin, 1993, ch. 7, pp. 131-159.

A. W. F. Edwards, A quick route to sums of powers, Amer. Math. Monthly 93 (1986) 451-455.
D. E. Knuth, Johann Faulhaber and sums of powers, Math. Comput. 203 (1993), 277-294.
Wolfdieter Lang, First 10 rows and Faulhaber triangle with rational entries and examples.
D. Yeliussizov, Permutation Statistics on Multisets, Ph.D. Dissertation, Computer Science, Kazakh-British Technical University, 2012. - _N. J. A. Sloane_, Jan 03 2013

Cf. A093557 (denominators).
Cf. A065551 and A065553 for Ira M. Gessel's and X. G. Viennot's version of Faulhaber triangle which is Edwards' Faulhaber triangle augmented with a first row and first column.
Cf. A027641, A027642, A093558, A093559, A103438.

a[m_, k_] := (-1)^(m-k)*Sum[ Binomial[2*m, m-k-j]*Binomial[m-k+j, j]*((m-k-j)/(m-k+j))*BernoulliB[m+k+j], {j, 0, m-k}]; Flatten[ Table[ Numerator[a[m, k]], {m, 1, 11}, {k, 0, m-1}]] (* Jean-François Alcover, Oct 25 2011 *)

T(n,k) = numerator((-1)^(n-k)*sum(j=0, n-k, binomial(2*n, n-k-j)*binomial(n-k+j,j)*(n-k-j)/(n-k+j) * bernfrac(n+k+j))); \\ Michel Marcus, Aug 03 2025

a(m, k) = numerator(A(m, k)) with recursion: A(m, 0)=1, A(m, k) = -(Sum_{j=0..k-1} binomial(m-j, 2*k+1-2*j)*A(m, j))/(m-k) if 0 <= k <= m-1, otherwise 0. From the Knuth 1993 reference, p. 288, eq.(*) with A^{(m)}_k = A(m, k).

A(m, k) = ((-1)^(m-k))*Sum_{j=0..m-k} binomial(2*m, m-k-j)*binomial(m-k+j, j)*((m-k-j)/(m-k+j))*Bernoulli(m+k+j). From the Knuth 1993 reference, p. 289, last eq. with A^{(m)}_k = A(m, k). Attributed to I. M. Gessel and X. G. Viennot (see A065551 for the 1989 reference). For Bernoulli numbers see A027641 with A027642.

More terms from Michel Marcus, Aug 03 2025

A093558 Triangle of numerators of coefficients of Faulhaber polynomials used for sums of even powers.

Original entry on oeis.org

1, 1, -1, 1, -1, 1, 1, -1, 1, -1, 1, -5, 17, -5, 5, 1, -5, 41, -236, 691, -691, 1, -7, 14, -22, 359, -7, 7, 1, -14, 77, -293, 1519, -1237, 3617, -3617, 1, -6, 217, -1129, 8487, -6583, 750167, -43867, 43867, 1, -5, 23, -470, 689, -28399, 1540967, -1254146, 174611, -174611

Offset: 2

Wolfdieter Lang, Apr 02 2004

The companion triangle with the denominators is A093559.

Sum_{k=1..n} k^(2*(m-1)) = (2*n+1)*Sum_{j=0..m-1} Fe(m,k)*(n*(n+1))^(m-1-j), m >= 2. Sums of even powers of the first n integers >0 as polynomials in u := n*(n+1) (falling powers of u). See bottom of p. 288 of the 1993 Knuth reference.

			Triangle begins:
  [1];
  [1,-1];
  [1,-1,1];
  [1,-1,1,-1];
  [1,-5,17,-5,5]
   ...
Numerators of:
  [1/6];
  [1/10,-1/30];
  [1/14,-1/14,1/42];
  [1/18,-1/9,1/10,-1/30];
  [1/22,-5/33,17/66,-5/22,5/66];
  ... (see Lang link)

Ivo Schneider, Johannes Faulhaber 1580-1635, Birkhäuser Verlag, Basel, Boston, Berlin, 1993, ch. 7, pp. 131-159.

A. Dzhumadil'daev and D. Yeliussizov, Power sums of binomial coefficients, Journal of Integer Sequences, 16 (2013), Article 13.1.4.
D. E. Knuth, Johann Faulhaber and sums of powers, Math. Comput. 203 (1993), 277-294.
Wolfdieter Lang, First 10 rows and triangle with rational entries.
D. Yeliussizov, Permutation Statistics on Multisets, Ph.D. Dissertation, Computer Science, Kazakh-British Technical University, 2012. [_N. J. A. Sloane_, Jan 03 2013]

Cf. A093556/A093557, A093559 (denominators).

a[m_, k_] := (-1)^(m-k)*Sum[Binomial[2*m, m-k-j]*Binomial[m-k+j, j]*((m-k-j)/(m-k+j))*BernoulliB[m+k+j], {j, 0, m-k}]; t[m_, k_] := (m-k)*a[m, k]/(2*m*(2*m-1)); Table[t[m, k] // Numerator, {m, 2, 12}, {k, 0, m-2}] // Flatten (* Jean-François Alcover, Mar 03 2014 *)

a(n, m) = numerator(Fe(m, k), with Fe(m, k):=(m-k)*A(m, k)/(2*m*(2*m-1)) with Faulhaber numbers A(m, k):=A093556(m, k)/A093557(m, k) in Knuth's version. From the bottom of p. 288 of the 1993 Knuth reference.

A093559 Triangle of denominators of coefficients of Faulhaber polynomials used for sums of even powers.

Original entry on oeis.org

6, 10, 30, 14, 14, 42, 18, 9, 10, 30, 22, 33, 66, 22, 66, 26, 26, 78, 273, 910, 2730, 30, 30, 15, 9, 90, 2, 6, 34, 51, 51, 51, 102, 51, 170, 510, 38, 19, 95, 95, 190, 57, 3990, 266, 798, 42, 14, 7, 21, 6, 66, 1386, 693, 110, 330, 46, 138, 46, 23, 230, 690, 345, 23, 230, 46

Offset: 2

Wolfdieter Lang, Apr 02 2004

The companion triangle with the numerators is A093558. See comment there.

			Triangle begins:
  [6];
  [10,30];
  [14,14,42];
  [18,9,10,30]; ...
Denominators of:
  [1/6];
  [1/10,-1/30];
  [1/14,-1/14,1/42];
  [1/18,-1/9,1/10,-1/30];
  ... (see W. Lang link in A093558.)

Ivo Schneider, Johannes Faulhaber 1580-1635, Birkhäuser Verlag, Basel, Boston, Berlin, 1993, ch. 7, pp. 131-159.

A. Dzhumadil'daev and D. Yeliussizov, Power sums of binomial coefficients, Journal of Integer Sequences, 16 (2013), Article 13.1.4.
D. E. Knuth, Johann Faulhaber and sums of powers, Math. Comput. 203 (1993), 277-294.
D. Yeliussizov, Permutation Statistics on Multisets, Ph.D. Dissertation, Computer Science, Kazakh-British Technical University, 2012. [_N. J. A. Sloane_, Jan 03 2013]

a[m_, k_] := (-1)^(m-k)*Sum[Binomial[2*m, m-k-j]*Binomial[m-k+j, j]*((m-k-j)/(m-k+j))*BernoulliB[m+k+j], {j, 0, m-k}]; t[m_, k_] := (m-k)*a[m, k]/(2*m*(2*m-1)); Table[t[m, k] // Denominator, {m, 2, 12}, {k, 0, m-2}] // Flatten (* Jean-François Alcover, Mar 03 2014 *)

a(n, m) = denominator(Fe(m, k), with Fe(m, k):=(m-k)*A(m, k)/(2*m*(2*m-1)) with Faulhaber numbers A(m, k):=A093556(m, k)/A093557(m, k) in Knuth's version. From the bottom of p. 288 of the 1993 Knuth reference.

A335951 Triangle read by rows. The numerators of the coefficients of the Faulhaber polynomials. T(n,k) for n >= 0 and 0 <= k <= n.

Original entry on oeis.org

1, 0, 1, 0, 0, 1, 0, 0, -1, 4, 0, 0, 1, -4, 6, 0, 0, -3, 12, -20, 16, 0, 0, 5, -20, 34, -32, 16, 0, 0, -691, 2764, -4720, 4592, -2800, 960, 0, 0, 105, -420, 718, -704, 448, -192, 48, 0, 0, -10851, 43404, -74220, 72912, -46880, 21120, -6720, 1280

Offset: 0

Peter Luschny, Jul 16 2020

There are many versions of Faulhaber's triangle: search the OEIS for his name.
Faulhaber's claim (in 1631) is: S_{2*m-1} = 1^(2*m-1) + 2^(2*m-1) + ... + n^(2*m-1) = F_m((n^2+2)/2). The first proof was given by Jacobi in 1834.
For the Faulhaber numbers see A354042 and A354043.

			The first few polynomials are:
  [0] 1;
  [1] x;
  [2] x^2;
  [3] (4*x - 1)*x^2*(1/3);
  [4] (6*x^2 - 4*x + 1)*x^2*(1/3);
  [5] (16*x^3 - 20*x^2 + 12*x - 3)*x^2*(1/5);
  [6] (16*x^4 - 32*x^3 + 34*x^2 - 20*x + 5)*x^2*(1/3);
  [7] (960*x^5 - 2800*x^4 + 4592*x^3 - 4720*x^2 + 2764*x - 691)*x^2*(1/105);
  [8] (48*x^6 - 192*x^5 + 448*x^4 - 704*x^3 + 718*x^2 - 420*x + 105)*x^2*(1/3);
  [9] (1280*x^7-6720*x^6+21120*x^5-46880*x^4+72912*x^3-74220*x^2+43404*x-10851)*x^2*(1/45);
Triangle starts:
  [0] 1;
  [1] 0, 1;
  [2] 0, 0,  1;
  [3] 0, 0, -1,     4;
  [4] 0, 0,  1,    -4,      6;
  [5] 0, 0, -3,     12,    -20,    16;
  [6] 0, 0,  5,    -20,     34,   -32,     16;
  [7] 0, 0, -691,   2764,  -4720,  4592,  -2800,  960;
  [8] 0, 0,  105,  -420,    718,  -704,    448,  -192,    48;
  [9] 0, 0, -10851, 43404, -74220, 72912, -46880, 21120, -6720, 1280;

Johann Faulhaber, Academia Algebra. Darinnen die miraculosische Inventiones zu den höchsten Cossen weiters continuirt und profitiert werden. Johann Ulrich Schönigs, Augsburg, 1631.

C. G. J. Jacobi, De usu legitimo formulae summatoriae Maclaurinianae, J. Reine Angew. Math., 12 (1834), 263-272.
Donald E. Knuth, Johann Faulhaber and sums of powers, arXiv:math/9207222 [math.CA], 1992; Math. Comp. 61 (1993), no. 203, 277-294.
Peter Luschny, Illustrating the Faulhaber polynomials for n = 1..7.

Cf. A335952 (polynomial denominators), A000012 (row sums of the polynomial coefficients).
Other representations of the Faulhaber polynomials include A093556/A093557, A162298/A162299, A220962/A220963.
Cf. A354042 (Faulhaber numbers), A354043.

FaulhaberPolynomial := proc(n) if n = 0 then return 1 fi;
expand((bernoulli(2*n, x+1) - bernoulli(2*n,1))/(2*n));
sort(simplify(expand(subs(x = (sqrt(8*x+1)-1)/2, %))), [x], ascending) end:
Trow := n -> seq(coeff(numer(FaulhaberPolynomial(n)), x, k), k=0..n):
seq(print(Trow(n)), n=0..9);

from math import lcm
from itertools import count, islice
from sympy import simplify,sqrt,bernoulli
from sympy.abc import x
def A335951_T(n,k):
    z = simplify((bernoulli(2*n,(sqrt(8*x+1)+1)/2)-bernoulli(2*n,1))/(2*n)).as_poly().all_coeffs()
    return z[n-k]*lcm(*(d.q for d in z))
def A335951_gen(): # generator of terms
    yield from (A335951_T(n,k) for n in count(0) for k in range(n+1))
A335951_list = list(islice(A335951_gen(),20)) # Chai Wah Wu, May 16 2022

def A335951Row(n):
    R. = PolynomialRing(QQ)
    if n == 0: return [1]
    b = expand((bernoulli_polynomial(x + 1, 2*n) -
                bernoulli_polynomial(1, 2*n))/(2*n))
    s = expand(b.subs(x = (sqrt(8*x+1)-1)/2))
    return numerator(s).list()
for n in range(10): print(A335951Row(n)) # Peter Luschny, May 17 2022

Let F_n(x) be the polynomial after substituting (sqrt(8*x + 1) - 1)/2 for x in b_n(x), where b_n(x) = (Bernoulli_{2*n}(x+1) - Bernoulli_{2*n}(1))/(2*n).
F_n(1) = 1 for all n >= 0.
T(n, k) = numerator([x^k] F_n(x)).

A065551 Triangle of Faulhaber numbers (numerators) read by rows.

Original entry on oeis.org

1, 0, 1, 0, -1, 1, 0, 1, -1, 1, 0, -3, 3, -1, 1, 0, 5, -5, 17, -2, 1, 0, -691, 691, -118, 41, -5, 1, 0, 35, -35, 359, -44, 14, -1, 1, 0, -3617, 3617, -1237, 1519, -293, 22, -7, 1, 0, 43867, -43867, 750167, -13166, 2829, -2258, 217, -4, 1, 0, -1222277, 1222277, -627073, 1540967, -198793, 689, -235, 46, -3, 1

Offset: 0

Wouter Meeussen, Dec 02 2001

From Wolfdieter Lang, Jun 25 2011: (Start)
In the Gessel and Viennot reference f(n,k) = a(n,k)/A065553(n,k), n>=0, k>=0.
(n+1)*f(n,k) = A(n+1,n-k), with Knuth's A(m,k) =
A093556(m,k)/A093557(m,k). See the Knuth reference given in A093556, and the W. Lang link. (End)

			Triangle begins:
{1},
{0, 1},
{0, -1, 1},
{0, 1, -1, 1},
{0, -3, 3, -1, 1},
{0, 5, -5, 17, -2, 1}.

Ira M. Gessel and X. G. Viennot, Determinants, paths and plane partitions, 1989, p. 27, eqn 12.2

Cf. A065553.

Cf. A103438.

sum(n>=0, k>=0, f(n, k)*t^k*x^(2*n+1)/(2*n+1)! ) is the expansion of (cosh(sqrt(1+4*t)*x/2)-cosh(x/2))/t/sinh(x/2).

a(n,k)=numerator(f(n,k)).

A385567 Triangle read by rows: T(n,k) is the numerator of A(n,k), such that A(n,k) satisfies the identity for sums of odd powers: Sum_{k=1..p} k^(2n-1) = 1/(2n) Sum_{k=0..n-1} A(n,k) * (p^2+p)^(n-k), for all integers p >= 1.

Original entry on oeis.org

1, 1, 1, 1, 0, -1, 1, -1, 0, 1, 1, -4, 2, 0, -1, 1, -5, 3, -3, 0, 5, 1, -4, 17, -10, 5, 0, -691, 1, -35, 287, -118, 691, -691, 0, 7, 1, -8, 112, -352, 718, -280, 140, 0, -3617, 1, -21, 66, -293, 4557, -3711, 10851, -10851, 0, 43867, 1, -40, 217, -4516, 2829, -26332, 750167, -438670, 219335, 0, -174611

Offset: 0

Kolosov Petro, Jul 31 2025

The companion triangle with the denominators is A386728.

Extension of A093556 with k in the range 0 <= k <= n, and n >= 0.

			Triangle begins:
---------------------------------------------------------------------------------
k =   0    1     2     3     4       5       6        7       8      9      10
---------------------------------------------------------------------------------
n=0:  1;
n=1:  1,   1;
n=2:  1,   0,  -1;
n=3:  1,  -1,   0,     1;
n=4:  1,  -4,   2,     0,   -1;
n=5:  1,  -5,   3,    -3,    0,      5;
n=6:  1,  -4,  17,   -10,    5,      0,   -691;
n=7:  1, -35, 287,  -118,  691,   -691,      0,       7;
n=8:  1,  -8, 112,  -352,  718,   -280,    140,       0,  -3617;
n=9:  1, -21,  66,  -293, 4557,  -3711,  10851,  -10851,      0, 43867;
n=10: 1, -40, 217, -4516, 2829, -26332, 750167, -438670, 219335,     0, -174611;
...

Donald E. Knuth, Johann Faulhaber and Sums of Powers, arXiv:9207222 [math.CA], 1992, see page 16.
Petro Kolosov, Faulhaber's coefficients: Examples, GitHub, 2025.
Petro Kolosov, Mathematica programs, GitHub, 2025.

Cf. A386728 (denominators).
Cf. A303675, A304330, A304334, A304336.
Cf. A093558/A093559, A335951/A335952, A093556/A093557.

FaulhaberCoefficient[n_, k_] := 0;
FaulhaberCoefficient[n_, k_] := (-1)^(n - k) * Sum[Binomial[2 n, n - k - j]* Binomial[n - k + j, j] * (n - k - j)/(n - k + j) * BernoulliB[n + k + j], {j, 0, n - k}] /; 0 <= k < n;
FaulhaberCoefficient[n_, k_] := BernoulliB[2 n] /; k == n;
Flatten[Table[Numerator[FaulhaberCoefficient[n, k]], {n, 0, 10}, {k, 0, n}]]

T(n,k) = numerator(if (k==n, bernfrac(2*n), if (kMichel Marcus, Aug 03 2025

A(n,k) = 0 if k>n or n<0
A(n,k) = (-1)^(n - k) * Sum_{j=0..n-k} binomial(2n, n - k - j) * binomial(n - k + j, j) * (n - k - j)/(n - k + j) * B_{n + k + j}, if 0 <= k < n;
A(n,k) = B_{2n}, if k = n;
T(n,k) = numerator(A(n,k)).

A386728 Triangle read by rows: T(n,k) is the denominator of A(n,k), such that A(n,k) satisfies the identity for sums of odd powers: Sum_{k=1..p} k^(2n-1) = 1/(2n) Sum_{k=0..n-1} A(n,k) * (p^2+p)^(n-k), for all integers p >= 1.

Original entry on oeis.org

1, 1, 6, 1, 1, 30, 1, 2, 1, 42, 1, 3, 3, 1, 30, 1, 2, 1, 2, 1, 66, 1, 1, 2, 1, 1, 1, 2730, 1, 6, 15, 3, 15, 30, 1, 6, 1, 1, 3, 3, 3, 1, 1, 1, 510, 1, 2, 1, 1, 5, 2, 5, 10, 1, 798, 1, 3, 2, 7, 1, 3, 42, 21, 21, 1, 330, 1, 2, 3, 2, 1, 6, 15, 3, 5, 10, 1, 138, 1

Offset: 0

Kolosov Petro, Jul 31 2025

The companion triangle with the numerators is A385567.

Extension of A093557 with k in the range 0 <= k <= n.

			Triangle begins:
  ---------------------------------------------------------
  k =   0  1   2   3    4    5    6   7    8    9    10
  ---------------------------------------------------------
  n=0:  1;
  n=1:  1, 6;
  n=2:  1, 1, 30;
  n=3:  1, 2,  1, 42;
  n=4:  1, 3,  3,  1, 30;
  n=5:  1, 2,  1,  2,  1, 66;
  n=6:  1, 1,  2,  1,  1,  1, 2730;
  n=7:  1, 6, 15,  3, 15, 30,    1,  6;
  n=8:  1, 1,  3,  3,  3,  1,    1,  1, 510;
  n=9:  1, 2,  1,  1,  5,  2,    5, 10,   1, 798;
  n=10: 1, 3,  2,  7,  1,  3,   42, 21,  21,   1, 330;
  ...

Donald E. Knuth, Johann Faulhaber and Sums of Powers, arXiv:9207222 [math.CA], 1992, see page 16.
Petro Kolosov, Faulhaber's coefficients: Examples, GitHub, 2025.
Petro Kolosov, Mathematica programs, GitHub, 2025.

Cf. A385567 (numerators).
Cf. A303675, A304330, A304334, A304336.
Cf. A093558/A093559, A335951/A335952, A093556/A093557.

FaulhaberCoefficient[n_, k_] := 0;
FaulhaberCoefficient[n_, k_] := (-1)^(n - k) * Sum[Binomial[2 n, n - k - j]* Binomial[n - k + j, j] * (n - k - j)/(n - k + j) * BernoulliB[n + k + j], {j, 0, n - k}] /; 0 <= k < n;
FaulhaberCoefficient[n_, k_] := BernoulliB[2 n] /; k == n;
Flatten[Table[Denominator[FaulhaberCoefficient[n, k]], {n, 0, 10}, {k, 0, n}]]

T(n,k) = denominator(if (k==n, bernfrac(2*n), if (kMichel Marcus, Aug 03 2025

A(n,k) = 0 if k>n or n<0;
A(n,k) = (-1)^(n - k) * Sum_{j=0..n-k} binomial(2n, n - k - j) * binomial(n - k + j, j) * (n - k - j)/(n - k + j) * B_{n + k + j}, if 0 <= k < n;
A(n,k) = B_{2n}, if k = n;
T(n,k) = denominator(A(n,k)).

A251926 The Faulhaber-Knuth a(0,n) sequence.

Original entry on oeis.org

2, 1, 1, 1, 1, 0, 0, 1, 37, -60, -5, 37, 174, -955, -10545, 38610, 176297, -322740, -205420, 4512655, 56820585, -104019264, -25907081, 94854194, 1141847218, -2090335775, -414239903275, 6066664425833, 85621405759989, -156743813184120, -4337631088920, 47644406040193, 1265208493396175131, -2316168508680582540, -192288633159406495

Offset: 4

Roman Witula, Dec 11 2014

sign

a(n) is equal to the remainder when dividing the polynomial T_n(x) by x^2 + x - 1. T_n(x) (in Z[x]) is the positive integer multiplicity of the modified Faulhaber polynomial T*_n(x), coefficients of which have GCD equal to 1. We have T*_n(x) = S(n;x)/x^2(x+1)^2 if n is odd, and T*_n(x) = S(n;x)/x(x+1)(2x+1) if n is even, n >= 4, where S(n;x) denotes the n-th Faulhaber polynomial, i.e., S(n;x) = 1/(n+1) sum{taken over i=0,1,...,n} Bin(n+1,i)Bern(i)x^(n+1-i), and Bern(i) denotes the i-th Bernoulli number with Bern(1)=1/2.

We note that every T_n(x) is a polynomial in the variable (x^2 + x - 1), for example T_7(x) = 3(x^2 + x - 1)^2 + 2(x^2 + x - 1) + 1. Furthermore, every T_n(x) is a polynomial in (x^2 + x + a) for each complex a. But only for a = -1 is the element a(n) also equal to the remainder when dividing S(n;x) by x^2 + x + a if n is odd and S(n;x)/(2x+1) by x^2 + x + a if n is even.

			We have: T_4(x) = 3x^2 + 3x - 1, T_4(x) - T_5(x) = x^2 + x, T_6(x) - T_7(x) = x^2 + x - 1, T_9(x) = (x^2 + x - 1)(2x^4 + 4x^3 - x^2 - 3x + 3) and T_15(x) - T_12(x) is divisible by (x^2 + x - 1), which implies a(0)=2, a(1)=1, a(2)=a(3), a(5)=0 and a(8)=a(11).

Edyta Hetmaniok, Piotr Lorenc, Mariusz Pleszczyński, and Roman Wituła, Iterated integrals of polynomials, Applied Mathematics and Computation, Volume 249, 15 December 2014, Pages 389-398.
D. E. Knuth, Johann Faulhaber and sums of powers, Math. Comput. 203 (1993), 277-294.
Piotr Lorenc, Jakub Jan Ludew, Mariusz Pleszczyński, Alicja Samulewicz, and Roman Wituła, Iterated integrals of Faulhaber polynomials and some properties of their roots, 2018.

Cf. A093556, A093557.

coeffFaulh[n_] := Module[{t, tab = {}, s, p, x},
  If[n < 4, Return["Give n greater than 3."]];
  t = Table[1, {n + 2}];
  Do[t[[i + 1]] = BernoulliB[i], {i, 1, n + 1}];
  t[[2]] = 1/2;
  s[m_, x_] := (Sum[Binomial[m + 1, i]t[[ i + 1]] x^(m + 1 - i),{i,0,m}])/(m + 1);
  Do[If[Mod[i, 2] == 0,
    p = PolynomialRemainder[FactorList[Factor[s[i, x]] (i + 1)/(x (x + 1) (2 x + 1))][[2,1]], -1 + x + x^2, x],
    p = PolynomialRemainder[FactorList[Factor[s[i, x]] (i + 1)/(x^2 (x + 1)^2)][[2,1]], -1 + x + x^2, x]];
   tab = Append[tab, p], {i, 4, n}];
  tab]

cp's OEIS Frontend

A196837 Coefficient table of numerator polynomials of o.g.f.s for partial sums of powers of positive integers.

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A093556 Triangle of numerators of coefficients of Faulhaber polynomials in Knuth's version.

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A093558 Triangle of numerators of coefficients of Faulhaber polynomials used for sums of even powers.

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A093559 Triangle of denominators of coefficients of Faulhaber polynomials used for sums of even powers.

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A335951 Triangle read by rows. The numerators of the coefficients of the Faulhaber polynomials. T(n,k) for n >= 0 and 0 <= k <= n.

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A065551 Triangle of Faulhaber numbers (numerators) read by rows.

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A385567 Triangle read by rows: T(n,k) is the numerator of A(n,k), such that A(n,k) satisfies the identity for sums of odd powers: Sum_{k=1..p} k^(2n-1) = 1/(2*n) * Sum_{k=0..n-1} A(n,k) * (p^2+p)^(n-k), for all integers p >= 1.

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A385567 Triangle read by rows: T(n,k) is the numerator of A(n,k), such that A(n,k) satisfies the identity for sums of odd powers: Sum_{k=1..p} k^(2n-1) = 1/(2n) Sum_{k=0..n-1} A(n,k) * (p^2+p)^(n-k), for all integers p >= 1.

A386728 Triangle read by rows: T(n,k) is the denominator of A(n,k), such that A(n,k) satisfies the identity for sums of odd powers: Sum_{k=1..p} k^(2n-1) = 1/(2n) Sum_{k=0..n-1} A(n,k) * (p^2+p)^(n-k), for all integers p >= 1.