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
Examples
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; ...
Links
- 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.
Crossrefs
Programs
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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}]] -
PARI
T(n,k) = numerator(if (k==n, bernfrac(2*n), if (k
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)).
Comments