A061684 Generalized Bell numbers: column 3 of A275043.
1, 1, 5, 64, 1613, 69026, 4566992, 437665649, 57903766797, 10193400044254, 2319001344297830, 665816738235745559, 236563125351122920088, 102303284135845463907107, 53093636013475924370369829, 32666276100771741793923209939, 23573762287735885858839134983437
Offset: 0
Keywords
Links
- Alois P. Heinz, Table of n, a(n) for n = 0..215
- J.-M. Sixdeniers, K. A. Penson and A. I. Solomon, Extended Bell and Stirling Numbers From Hypergeometric Exponentiation, J. Integer Seqs. Vol. 4 (2001), #01.1.4.
Crossrefs
Programs
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Maple
b:= proc(n) option remember; `if`(n=0, 1, add(b(n-i)*binomial(n-1, i-1)/i!^2, i=1..n)) end: a:= n-> b(n)*n!^2: seq(a(n), n=0..20); # Alois P. Heinz, May 11 2016 -
Mathematica
b[n_] := b[n] = If[n==0, 1, Sum[b[n-i]*Binomial[n-1, i-1]/i!^2, {i, 1, n}]]; a[n_] := b[n]*n!^2; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Mar 14 2017, after Alois P. Heinz *) -
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 */
Formula
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]
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
Extensions
More terms from Karol A. Penson, Sep 10 2001