A061690 Generalized Stirling numbers.

0, 0, 6, 72, 650, 5400, 43757, 353192, 2862330, 23352300, 191891117, 1587587760, 13215894133, 110619113423, 930376519256, 7858437064232, 66627124896218, 566791391339532, 4836144006188165, 41375938305568772, 354859541163656045

Offset

1,3

Links

Table of n, a(n) for n=1..21.

J.-M. Sixdeniers, K. A. Penson and A. I. Solomon, Extended Bell and Stirling Numbers From Hypergeometric Exponentiation, J. Integer Seqs. Vol. 4 (2001), #01.1.4.

Formula

a(n) = s(n, k) = Sum_{pi} n!/(k(1)! * 1!^k(1) * k(2)! * 2!^k(2) * ... * k(n)! * n!^k(n)) * (n!/(1!^k(1) * 2!^k(2) * ... * n!^k(n)))^L, where pi runs through all partitions k(1) +2 * k(2)+...+n * k(n) = n such that k = k(1)+k(2)+...+k(n), with k = 3 and L = 1.

Sum_{n>=1} a(n) * x^n / (n!)^2 = (BesselI(0,2*sqrt(x)) - 1)^3 / 6. - Ilya Gutkovskiy, Mar 04 2021

Prog

(PARI) a(n) = {n!^2*polcoef((besseli(0, 2*x + O(x^(2*n+1))) - 1)^3, 2*n)/6} \\ Andrew Howroyd, Mar 04 2021

Crossrefs

Third column of A061691.

Sequence in context: A283095 A281774 A036292 * A133678 A005548 A059410

Adjacent sequences: A061687 A061688 A061689 * A061691 A061692 A061693

Keyword

nonn

Author

N. J. A. Sloane, Jun 18 2001

Extensions

Formula and more terms from Vladeta Jovovic, Dec 09 2001

Status

approved