A335952 -id:A335952 - cp's OEIS frontend

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

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)).

A354042 Triangle read by rows. The Faulhaber numbers. F(0, k) = 1 and otherwise F(n, k) = (n + 1)!(-1)^(k+1)Sum_{j=0..floor((k-1)/2)} C(2k-2j, k+1)C(2n+1, 2j+1) Bernoulli(2n-2j) / (k - j).

Original entry on oeis.org

1, 0, 1, 0, -1, 2, 0, 4, -8, 6, 0, -36, 72, -60, 24, 0, 600, -1200, 1020, -480, 120, 0, -16584, 33168, -28320, 13776, -4200, 720, 0, 705600, -1411200, 1206240, -591360, 188160, -40320, 5040, 0, -43751232, 87502464, -74813760, 36747648, -11813760, 2661120, -423360, 40320

Offset: 0

Peter Luschny, May 17 2022

I. Gessel and X. Viennot call the rational numbers F(n, k)/(n + 1)! 'Faulhaber numbers'. However, for our purposes it is more convenient to define the integers F(n, k). For the Faulhaber polynomials see A335951/A335952.
Let S(r, m) = Sum_{k=0..m} k^r, with 0^0 = 1 and S(0, m) = m + 1. Faulhaber's theorem (the sums of powers formula) is:
  S(2*n+1, m) = (1/(n+1)!)*(1/2)*Sum_{k=0..n} F(n, k)*(m*(m + 1))^(k + 1).
Gessel and Viennot give two combinatorial interpretations for the Faulhaber numbers, for this see A354043.

			Triangle starts:
0: 1
1: 0,         1
2: 0,        -1,        2
3: 0,         4,       -8,         6
4: 0,       -36,       72,       -60,       24
5: 0,       600,    -1200,      1020,     -480,       120
6: 0,    -16584,    33168,    -28320,    13776,     -4200,     720
7: 0,    705600, -1411200,   1206240,  -591360,    188160,  -40320,    5040
8: 0, -43751232, 87502464, -74813760, 36747648, -11813760, 2661120, -423360, 40320
.
Let n = 4 and m = 3, then S(2*n + 1, m) = S(9, 3) = 20196. Faulhaber's formula gives this as (0*12 + (-36)*144 + 72*1728 + (-60)*20736 + 24*248832) / (2*120).

I. M. Gessel and X. G. Viennot, Determinants, Paths, and Plane Partitions, 1989 preprint.

Cf. A335951/A335952, A000367/A002445, A354043, A263445.

F := (n, k) -> ifelse(n = 0, 1, (n + 1)!*(-1)^(k + 1)*add(binomial(2*k - 2*j, k + 1)*binomial(2*n + 1, 2*j + 1)*bernoulli(2*n - 2*j) / (k - j), j = 0..(k - 1)/2)): for n from 0 to 8 do seq(F(n, k), k = 0..n) od;

F(n,1) = (2*n +1)*Bernoulli(2*n)*(n+1)! for n >= 1.
F(n,2) = -(4*n+2)*Bernoulli(2*n)*(n+1)! for n >= 2.
F(n,3) = ((10*n+5)*Bernoulli(2*n) + binomial(2*n+1,3)*Bernoulli(2*n-2)/2)*(n+1)! for n >= 3.

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)).

cp's OEIS Frontend

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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A354042 Triangle read by rows. The Faulhaber numbers. F(0, k) = 1 and otherwise F(n, k) = (n + 1)!*(-1)^(k+1)*Sum_{j=0..floor((k-1)/2)} C(2*k-2*j, k+1)*C(2*n+1, 2*j+1) * Bernoulli(2*n-2*j) / (k - j).

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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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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/(2*n) * Sum_{k=0..n-1} A(n,k) * (p^2+p)^(n-k), for all integers p >= 1.

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A354042 Triangle read by rows. The Faulhaber numbers. F(0, k) = 1 and otherwise F(n, k) = (n + 1)!(-1)^(k+1)Sum_{j=0..floor((k-1)/2)} C(2k-2j, k+1)C(2n+1, 2j+1) Bernoulli(2n-2j) / (k - j).

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.