A384089 a(n) = [x^n] Product_{k=0..n-1} (1 + k*x)^n.

1, 0, 1, 63, 7206, 1357300, 384271700, 153027592116, 81648987014364, 56259916067074896, 48646018448463951450, 51584263505394472459750, 65833976467770842558152992, 99553004175105699906002335098, 176031670802373999913671973955080, 359870756416991348769957239299854000

Offset: 0

Author

Seiichi Manyama, May 19 2025

Keywords

• nonn

Crossrefs

Cf. A342111, A384018, A384029.

Cf. A351507.

Programs

Mathematica

Table[SeriesCoefficient[Product[(1 + k*x)^n, {k, 0, n-1}], {x, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, May 19 2025 *)

PARI

a(n) = polcoef(prod(k=0, n-1, 1+k*x)^n, n);

Formula

a(n) = Sum_{0 <= x_1, x_2,..., x_n <= n and x_1 + x_2 + ... + x_n = (n-1)*n} Product_{k=1..n} |Stirling1(n,x_k)|.

a(n) ~ exp(n - 5/3) * n^(2*n+1) / (sqrt(Pi) * n^(3/2) * 2^(n + 1/2)). - Vaclav Kotesovec, May 19 2025