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.

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

0,12

Comments

The companion triangle with the denominators is A386728.

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

Links

Table of n, a(n) for n=0..65.

Donald E. Knuth, Johann Faulhaber and Sums of Powers, arXiv:math/9207222 [math.CA], 1992, see page 16.

Petro Kolosov, Faulhaber's coefficients: Examples, GitHub, 2025.

Petro Kolosov, Mathematica programs, GitHub, 2025.

Formula

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)).

Example

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;
...

Mathematica

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}]]

Prog

(PARI) T(n, k) = numerator(if (k==n, bernfrac(2*n), if (k<n, (-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

Crossrefs

Cf. A386728 (denominators).

Cf. A303675, A304330, A304334, A304336.

Cf. A093558/A093559, A335951/A335952, A093556/A093557.

Sequence in context: A137986 A330768 A093486 * A259618 A377036 A354681

Adjacent sequences: A385564 A385565 A385566 * A385568 A385569 A385570

Keyword

sign,tabl,frac,easy

Author

Petro Kolosov, Jul 31 2025

Status

approved