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%I #17 Apr 17 2022 14:11:28
%S 0,4,27,172,1125,7591,52479,369580,2640465,19082629,139207959,
%T 1023462199,7574172879,56369211679,421563478527,3166149812140,
%U 23868662788809,180538738842217,1369635435497367,10418413517675797
%N Generalized Bell numbers.
%H J.-M. Sixdeniers, K. A. Penson and A. I. Solomon, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL4/SIXDENIERS/bell.html">Extended Bell and Stirling Numbers From Hypergeometric Exponentiation</a>, J. Integer Seqs. Vol. 4 (2001), #01.1.4.
%F a(n) = A000172(n)/2-1. - _Vladeta Jovovic_, Apr 23 2003
%F Sum_{n>=1} a(n) * x^n / (n!)^3 = (1/2) * ( Sum_{n>=1} x^n / (n!)^3 )^2. - _Ilya Gutkovskiy_, Mar 04 2021
%t a[n_] := Sum[Binomial[n, k]^3, {k, 0, n}]/2 - 1;
%t a[n_] := n^2*HypergeometricPFQ[{1 - n, 1 - n, 1 - n}, {2, 2}, -1] - 1;
%t Table[a[n], {n, 1, 20}] (* _Peter Luschny_, Apr 17 2022 *)
%K nonn
%O 1,2
%A _N. J. A. Sloane_, Jun 19 2001
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