A061693 Generalized Bell numbers.

0, 4, 27, 172, 1125, 7591, 52479, 369580, 2640465, 19082629, 139207959, 1023462199, 7574172879, 56369211679, 421563478527, 3166149812140, 23868662788809, 180538738842217, 1369635435497367, 10418413517675797

Offset: 1

Author

N. J. A. Sloane, Jun 19 2001

Keywords

• nonn

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.

Programs

Mathematica

a[n_] := Sum[Binomial[n, k]^3, {k, 0, n}]/2 - 1;
a[n_] := n^2*HypergeometricPFQ[{1 - n, 1 - n, 1 - n}, {2, 2}, -1] - 1;
Table[a[n], {n, 1, 20}] (* Peter Luschny, Apr 17 2022 *)

Formula

a(n) = A000172(n)/2-1. - Vladeta Jovovic, Apr 23 2003

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