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A061690
Generalized Stirling numbers.
1
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
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
KEYWORD
nonn
AUTHOR
N. J. A. Sloane, Jun 18 2001
EXTENSIONS
Formula and more terms from Vladeta Jovovic, Dec 09 2001
STATUS
approved