This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A385567 #56 Aug 07 2025 21:53:43
%S A385567 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,
%T A385567 1,-35,287,-118,691,-691,0,7,1,-8,112,-352,718,-280,140,0,-3617,1,-21,
%U A385567 66,-293,4557,-3711,10851,-10851,0,43867,1,-40,217,-4516,2829,-26332,750167,-438670,219335,0,-174611
%N 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.
%C A385567 The companion triangle with the denominators is A386728.
%C A385567 Extension of A093556 with k in the range 0 <= k <= n, and n >= 0.
%H A385567 Donald E. Knuth, <a href="https://arxiv.org/abs/math/9207222">Johann Faulhaber and Sums of Powers</a>, arXiv:9207222 [math.CA], 1992, see page 16.
%H A385567 Petro Kolosov, <a href="https://kolosovpetro.github.io/pdf/faulhabers-coefficients-examples.pdf">Faulhaber's coefficients: Examples</a>, GitHub, 2025.
%H A385567 Petro Kolosov, <a href="https://github.com/kolosovpetro/faulhabers-coefficients-examples/tree/main/mathematica">Mathematica programs</a>, GitHub, 2025.
%F A385567 A(n,k) = 0 if k>n or n<0
%F A385567 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;
%F A385567 A(n,k) = B_{2n}, if k = n;
%F A385567 T(n,k) = numerator(A(n,k)).
%e A385567 Triangle begins:
%e A385567 ---------------------------------------------------------------------------------
%e A385567 k = 0 1 2 3 4 5 6 7 8 9 10
%e A385567 ---------------------------------------------------------------------------------
%e A385567 n=0: 1;
%e A385567 n=1: 1, 1;
%e A385567 n=2: 1, 0, -1;
%e A385567 n=3: 1, -1, 0, 1;
%e A385567 n=4: 1, -4, 2, 0, -1;
%e A385567 n=5: 1, -5, 3, -3, 0, 5;
%e A385567 n=6: 1, -4, 17, -10, 5, 0, -691;
%e A385567 n=7: 1, -35, 287, -118, 691, -691, 0, 7;
%e A385567 n=8: 1, -8, 112, -352, 718, -280, 140, 0, -3617;
%e A385567 n=9: 1, -21, 66, -293, 4557, -3711, 10851, -10851, 0, 43867;
%e A385567 n=10: 1, -40, 217, -4516, 2829, -26332, 750167, -438670, 219335, 0, -174611;
%e A385567 ...
%t A385567 FaulhaberCoefficient[n_, k_] := 0;
%t A385567 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;
%t A385567 FaulhaberCoefficient[n_, k_] := BernoulliB[2 n] /; k == n;
%t A385567 Flatten[Table[Numerator[FaulhaberCoefficient[n, k]], {n, 0, 10}, {k, 0, n}]]
%o A385567 (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
%Y A385567 Cf. A386728 (denominators).
%Y A385567 Cf. A303675, A304330, A304334, A304336.
%Y A385567 Cf. A093558/A093559, A335951/A335952, A093556/A093557.
%K A385567 sign,tabl,frac,easy
%O A385567 0,12
%A A385567 _Kolosov Petro_, Jul 31 2025