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A061686 Generalized Bell numbers: column 5 of A275043.

%I A061686 #29 Jan 17 2025 11:21:42
%S A061686 1,1,17,1540,461105,350813126,573843627152,1797582928354025,
%T A061686 9904754169831094065,89944005095677792967482,
%U A061686 1278494002506675052860358142,27281796399886236251265603339575,844252087185585895268923657508727440,36800471170748991972750857754287551544147
%N A061686 Generalized Bell numbers: column 5 of A275043.
%H A061686 Vincenzo Librandi, <a href="/A061686/b061686.txt">Table of n, a(n) for n = 0..100</a>
%H A061686 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 A061686 Sum_{n>=0} a(n) * x^n / (n!)^5 = exp(Sum_{n>=1} x^n / (n!)^5). - _Ilya Gutkovskiy_, Jul 17 2020
%p A061686 a:= proc(n) option remember; `if`(n=0, 1,
%p A061686       add(binomial(n, k)^5*(n-k)*a(k)/n, k=0..n-1))
%p A061686     end:
%p A061686 seq(a(n), n=0..15); # _Alois P. Heinz_, Nov 07 2008
%t A061686 a[n_] := a[n] = If[n == 0, 1, Sum[Binomial[n, k]^5*(n-k)*a[k]/n, {k, 0, n-1}]]; Table[a[n], {n, 0, 15}] (* _Jean-François Alcover_, Mar 24 2014, after _Alois P. Heinz_ *)
%o A061686 (PARI) a61686=[1];A061686(n)={n>1||return(1);#a61686<n&&a61686=concat(a61686,vector(n-#a61686)); a61686[n]&&return(a61686[n]); a61686[n]=sum(k=0,n-1,binomial(n,k)^5*(n-k)*A061686(k))/n} \\ _M. F. Hasler_, May 11 2015
%Y A061686 Column k=5 of A275043.
%K A061686 nonn
%O A061686 0,3
%A A061686 _N. J. A. Sloane_, Jun 18 2001
%E A061686 More terms from _Alois P. Heinz_, Nov 07 2008