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A061685 Generalized Bell numbers: column 4 of A275043.

%I A061685 #23 Jan 17 2025 11:21:08
%S A061685 1,1,9,298,25097,4383626,1394519922,738298190981,608765840524809,
%T A061685 742996254490626106,1289282092211451157634,3078466688415490018129781,
%U A061685 9844321075186192301310239858,41209705023068976933023104392293,221473347301087557264532943397984133
%N A061685 Generalized Bell numbers: column 4 of A275043.
%H A061685 Alois P. Heinz, <a href="/A061685/b061685.txt">Table of n, a(n) for n = 0..167</a>
%H A061685 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 A061685 a(n) = 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, with L = 3.
%F A061685 a(0) = 1; a(n) = (1/n) * Sum_{k=0..n-1} binomial(n,k)^4 * (n-k) * a(k). - _Ilya Gutkovskiy_, Jul 12 2020
%p A061685 b:= proc(n) option remember; `if`(n=0, 1,
%p A061685       add(b(n-i)*binomial(n-1, i-1)/i!^3, i=1..n))
%p A061685     end:
%p A061685 a:= n-> b(n)*n!^3:
%p A061685 seq(a(n), n=0..20);  # _Alois P. Heinz_, May 11 2016
%t A061685 b[n_] := b[n] = If[n==0, 1, Sum[b[n-i]*Binomial[n-1, i-1]/i!^3, {i, n}]];
%t A061685 a[n_] := b[n]*n!^3;
%t A061685 Table[a[n], {n, 0, 20}] (* _Jean-François Alcover_, Mar 15 2017, after _Alois P. Heinz_ *)
%Y A061685 Column k=4 of A275043.
%K A061685 nonn
%O A061685 0,3
%A A061685 _N. J. A. Sloane_, Jun 18 2001
%E A061685 Formula and more terms from _Vladeta Jovovic_, Dec 09 2001