A061694 Generalized Bell numbers.

0, 0, 36, 864, 17500, 351000, 7197169, 151633440, 3275925804, 72315234000, 1625547144199, 37102497859152, 857909644412275, 20059247889751161, 473562712831103536, 11274693857547716640, 270435401233629732940

Offset: 1

Author

N. J. A. Sloane, Jun 19 2001

Keywords

• nonn

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..200
• Vaclav Kotesovec, Recurrence (of order 6)
• 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.

Programs

Mathematica

Table[Sum[Sum[(n!/(i!*j!*(n-i-j)!))^3/6,{i,1,n-j-1}],{j,1,n}],{n,1,20}] (* Vaclav Kotesovec, Mar 14 2014 *)

Formula

a(n) = 1/6*Sum_{i+j+k=n, i, j, k>0} (n!/(i!*j!*k!))^3. - Vladeta Jovovic, Apr 23 2003

a(n) ~ 3^(3*n+1) / (8*Pi^2*n^2). - Vaclav Kotesovec, Mar 14 2014

Sum_{n>=1} a(n) * x^n / (n!)^3 = (1/6) * ( Sum_{n>=1} x^n / (n!)^3 )^3. - Ilya Gutkovskiy, Mar 04 2021

Extensions

More terms from Vladeta Jovovic, Apr 23 2003