A383862 a(n) = [x^n] Product_{k=0..n} 1/(1 - k*x)^3.

1, 3, 48, 1386, 58278, 3225915, 221726711, 18216234288, 1741626159966, 189977753488050, 23285057201978520, 3168272346322892094, 473878954663846060735, 77281168674525142984020, 13647787698908399220563400, 2594721838238358445753776000, 528401900314147344955336365822

Offset: 0

Author

Seiichi Manyama, May 17 2025

Keywords

• nonn

Crossrefs

Cf. A007820, A350376, A384022, A384060.

Cf. A384012.

Programs

Mathematica

Table[SeriesCoefficient[Product[1/(1-k*x)^3, {k, 1, n}], {x, 0, n}], {n, 0, 16}] (* Vaclav Kotesovec, May 18 2025 *)
(* or *)
Table[Sum[StirlingS2[i + n, n] * StirlingS2[j + n, n] * StirlingS2[2*n - i - j, n], {i, 0, n}, {j, 0, n-i}], {n, 0, 16}] (* Vaclav Kotesovec, May 18 2025 *)

PARI

a(n) = sum(i=0, n, sum(j=0, n-i, stirling(i+n, n, 2)*stirling(j+n, n, 2)*stirling(2*n-i-j, n, 2)));

Formula

a(n) = Sum_{i, j, k>=0 and i+j+k=n} Stirling2(i+n,n) * Stirling2(j+n,n) * Stirling2(k+n,n).

a(n) ~ 2^(8*n + 3/2) * n^(n - 1/2) / (sqrt(Pi*(1-w)) * exp(n) * 3^(3*n + 3/2) * (4 - 3*w)^n * w^(3*n + 1)), where w = -LambertW(-4*exp(-4/3)/3) = 0.727473355414332993149219573314579663... - Vaclav Kotesovec, May 18 2025