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A061684 Generalized Bell numbers: column 3 of A275043.

%I A061684 #30 Jan 17 2025 11:20:35
%S A061684 1,1,5,64,1613,69026,4566992,437665649,57903766797,10193400044254,
%T A061684 2319001344297830,665816738235745559,236563125351122920088,
%U A061684 102303284135845463907107,53093636013475924370369829,32666276100771741793923209939,23573762287735885858839134983437
%N A061684 Generalized Bell numbers: column 3 of A275043.
%H A061684 Alois P. Heinz, <a href="/A061684/b061684.txt">Table of n, a(n) for n = 0..215</a>
%H A061684 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 A061684 G.f.: Sum_{n>=0} a(n)*x^n/n!^3 = exp( Sum_{n>=1} x^n/n!^3 ). [_Paul D. Hanna_, Mar 15 2012]
%F A061684 a(0) = 1; a(n) = (1/n) * Sum_{k=0..n-1} binomial(n,k)^3 * (n-k) * a(k). - _Ilya Gutkovskiy_, Jul 12 2020
%p A061684 b:= proc(n) option remember; `if`(n=0, 1,
%p A061684       add(b(n-i)*binomial(n-1, i-1)/i!^2, i=1..n))
%p A061684     end:
%p A061684 a:= n-> b(n)*n!^2:
%p A061684 seq(a(n), n=0..20);  # _Alois P. Heinz_, May 11 2016
%t A061684 b[n_] := b[n] = If[n==0, 1, Sum[b[n-i]*Binomial[n-1, i-1]/i!^2, {i, 1, n}]];
%t A061684 a[n_] := b[n]*n!^2;
%t A061684 Table[a[n], {n, 0, 20}] (* _Jean-François Alcover_, Mar 14 2017, after _Alois P. Heinz_ *)
%o A061684 (PARI) {a(n)=n!^3*polcoeff(exp(sum(m=1, n, x^m/m!^3)+x*O(x^n)), n)} /* _Paul D. Hanna_, Mar 15 2012 */
%Y A061684 Probably A061698 from the same paper is an erroneous version of this sequence. - _Les Reid_, Jan 01 2011
%Y A061684 Column k=3 of A275043.
%Y A061684 Row sums of A061692.
%K A061684 nonn
%O A061684 0,3
%A A061684 _N. J. A. Sloane_, Jun 18 2001
%E A061684 More terms from _Karol A. Penson_, Sep 10 2001