A383051 a(n) is the n-th term of the inverse Stirling transform of j-> (j+1)^n.

1, 2, 5, -1, -116, 984, 16400, -788418, 5474016, 941115360, -51647682648, -264087895512, 244846563852864, -16953959408998080, -436871956049596800, 219647419965976413744, -20283048895473275917824, -877465277974899660349440, 545297904370739513319183360

Offset: 0

Author

Seiichi Manyama, Apr 14 2025

Keywords

• sign

Links

• Christian G. Bower, PARI programs for transforms, 2007.
• N. J. A. Sloane, Maple programs for transforms, 2001-2020.

Crossrefs

Main diagonal of A383049.

Cf. A320082, A367820.

Programs

PARI

a(n) = sum(k=0, n, (k+1)^n*stirling(n, k, 1));

Formula

a(n) = Sum_{k=0..n} (k+1)^n * Stirling1(n,k).

a(n) = n! * [x^n] Sum_{k>=0} (k+1)^n * log(1+x)^k / k!.

a(n) = n! * [x^n] (1+x) * Sum_{k=0..n} Stirling2(n+1,k+1) * log(1+x)^k.