A061685 Generalized Bell numbers: column 4 of A275043.

1, 1, 9, 298, 25097, 4383626, 1394519922, 738298190981, 608765840524809, 742996254490626106, 1289282092211451157634, 3078466688415490018129781, 9844321075186192301310239858, 41209705023068976933023104392293, 221473347301087557264532943397984133

Offset: 0

Author

N. J. A. Sloane, Jun 18 2001

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..167
• 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

Column k=4 of A275043.

Programs

Maple

b:= proc(n) option remember; `if`(n=0, 1,
      add(b(n-i)*binomial(n-1, i-1)/i!^3, i=1..n))
    end:
a:= n-> b(n)*n!^3:
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!^3, {i, n}]];
a[n_] := b[n]*n!^3;
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Mar 15 2017, after Alois P. Heinz *)

Formula

a(n) = Sum_{pi} n!/(k(1)! * 1!^k(1) * k(2)! * 2!^k(2) * ... * k(n)! * n!^k(n)) * (n!/(1!^k(1) * 2!^k(2) * ... * n!^k(n)))^L, where pi runs through all partitions k(1) + 2 * k( 2) + ... + n * k(n) = n, with L = 3.

a(0) = 1; a(n) = (1/n) * Sum_{k=0..n-1} binomial(n,k)^4 * (n-k) * a(k). - Ilya Gutkovskiy, Jul 12 2020

Extensions

Formula and more terms from Vladeta Jovovic, Dec 09 2001