A061684 Generalized Bell numbers.

1, 1, 5, 64, 1613, 69026, 4566992, 437665649, 57903766797, 10193400044254, 2319001344297830, 665816738235745559, 236563125351122920088, 102303284135845463907107, 53093636013475924370369829, 32666276100771741793923209939, 23573762287735885858839134983437

Offset

0,3

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.

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

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 *)

Prog

(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 */

Crossrefs

Probably A061698 from the same paper is an erroneous version of this sequence. - Les Reid, Jan 01 2011

Column k=3 of A275043.

Row sums of A061692.

Sequence in context: A192558 A179156 A196304 * A061698 A351020 A238631

Adjacent sequences: A061681 A061682 A061683 * A061685 A061686 A061687

Keyword

nonn

Author

N. J. A. Sloane, Jun 18 2001

Extensions

More terms from Karol A. Penson, Sep 10 2001

Status

approved