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A061684 Generalized Bell numbers. 12

%I #28 Jul 12 2020 12:17:47

%S 1,1,5,64,1613,69026,4566992,437665649,57903766797,10193400044254,

%T 2319001344297830,665816738235745559,236563125351122920088,

%U 102303284135845463907107,53093636013475924370369829,32666276100771741793923209939,23573762287735885858839134983437

%N Generalized Bell numbers.

%H Alois P. Heinz, <a href="/A061684/b061684.txt">Table of n, a(n) for n = 0..215</a>

%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 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 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 b:= proc(n) option remember; `if`(n=0, 1,

%p add(b(n-i)*binomial(n-1, i-1)/i!^2, i=1..n))

%p end:

%p a:= n-> b(n)*n!^2:

%p seq(a(n), n=0..20); # _Alois P. Heinz_, May 11 2016

%t b[n_] := b[n] = If[n==0, 1, Sum[b[n-i]*Binomial[n-1, i-1]/i!^2, {i, 1, n}]];

%t a[n_] := b[n]*n!^2;

%t Table[a[n], {n, 0, 20}] (* _Jean-François Alcover_, Mar 14 2017, after _Alois P. Heinz_ *)

%o (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 Probably A061698 from the same paper is an erroneous version of this sequence. - _Les Reid_, Jan 01 2011

%Y Column k=3 of A275043.

%Y Row sums of A061692.

%K nonn

%O 0,3

%A _N. J. A. Sloane_, Jun 18 2001

%E More terms from _Karol A. Penson_, Sep 10 2001

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Last modified March 28 10:55 EDT 2024. Contains 371241 sequences. (Running on oeis4.)