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 A386728 #49 Aug 07 2025 21:53:49
%S A386728 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,
%T A386728 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,
%U A386728 3,42,21,21,1,330,1,2,3,2,1,6,15,3,5,10,1,138,1
%N 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.
%C A386728 The companion triangle with the numerators is A385567.
%C A386728 Extension of A093557 with k in the range 0 <= k <= n.
%H A386728 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 A386728 Petro Kolosov, <a href="https://kolosovpetro.github.io/pdf/faulhabers-coefficients-examples.pdf">Faulhaber's coefficients: Examples</a>, GitHub, 2025.
%H A386728 Petro Kolosov, <a href="https://github.com/kolosovpetro/faulhabers-coefficients-examples/tree/main/mathematica">Mathematica programs</a>, GitHub, 2025.
%F A386728 A(n,k) = 0 if k>n or n<0;
%F A386728 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 A386728 A(n,k) = B_{2n}, if k = n;
%F A386728 T(n,k) = denominator(A(n,k)).
%e A386728 Triangle begins:
%e A386728 ---------------------------------------------------------
%e A386728 k = 0 1 2 3 4 5 6 7 8 9 10
%e A386728 ---------------------------------------------------------
%e A386728 n=0: 1;
%e A386728 n=1: 1, 6;
%e A386728 n=2: 1, 1, 30;
%e A386728 n=3: 1, 2, 1, 42;
%e A386728 n=4: 1, 3, 3, 1, 30;
%e A386728 n=5: 1, 2, 1, 2, 1, 66;
%e A386728 n=6: 1, 1, 2, 1, 1, 1, 2730;
%e A386728 n=7: 1, 6, 15, 3, 15, 30, 1, 6;
%e A386728 n=8: 1, 1, 3, 3, 3, 1, 1, 1, 510;
%e A386728 n=9: 1, 2, 1, 1, 5, 2, 5, 10, 1, 798;
%e A386728 n=10: 1, 3, 2, 7, 1, 3, 42, 21, 21, 1, 330;
%e A386728 ...
%t A386728 FaulhaberCoefficient[n_, k_] := 0;
%t A386728 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 A386728 FaulhaberCoefficient[n_, k_] := BernoulliB[2 n] /; k == n;
%t A386728 Flatten[Table[Denominator[FaulhaberCoefficient[n, k]], {n, 0, 10}, {k, 0, n}]]
%o A386728 (PARI) T(n,k) = denominator(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 A386728 Cf. A385567 (numerators).
%Y A386728 Cf. A303675, A304330, A304334, A304336.
%Y A386728 Cf. A093558/A093559, A335951/A335952, A093556/A093557.
%K A386728 nonn,tabl,frac,easy
%O A386728 0,3
%A A386728 _Kolosov Petro_, Jul 31 2025