A382840 a(n) = Sum_{k=0..n} binomial(n+k-1,k) * Stirling1(n,k) * k!.

1, 1, 4, 30, 316, 4290, 71268, 1400112, 31750416, 816215760, 23455342560, 745073660496, 25924233481056, 980518650296640, 40054724743501440, 1757539560656401920, 82439565962427760896, 4116529729771939393920, 218017561353648160158720, 12206586491422209675532800

Offset: 0

Author

Ilya Gutkovskiy, Apr 06 2025

Keywords

• nonn

Crossrefs

Cf. A006252, A305919, A308565, A317280, A354120, A354121, A382830.

Programs

Mathematica

Table[Sum[Binomial[n + k - 1, k] StirlingS1[n, k] k!, {k, 0, n}], {n, 0, 19}]
Table[n! SeriesCoefficient[1/(1 - Log[1 + x])^n, {x, 0, n}], {n, 0, 19}]

Formula

a(n) = n! * [x^n] 1 / (1 - log(1 + x))^n.

a(n) ~ n^n / (sqrt(1 + LambertW(1)) * 2^n * exp(n) * (cosh(LambertW(1)) - 1)^n). - Vaclav Kotesovec, Apr 07 2025