A070531 Generalized Bell numbers B_{4,3}.

1, 73, 16333, 8030353, 7209986401, 10541813012041, 23227377813664333, 72925401604382826913, 312727862321385812968033, 1772004571987390827615327241, 12917715377912025572750844722221, 118521774439119390334062953438350513, 1343761301099219856651740487814621053313

Offset: 1

Author

Karol A. Penson, May 02 2002

Keywords

• nonn

Links

• G. C. Greubel, Table of n, a(n) for n = 1..200
• P. Blasiak, Karol A. Penson and A. I. Solomon, The general boson normal ordering problem, arXiv:quant-ph/0402027, 2004.
• P. Blasiak, Karol A. Penson and A. I. Solomon, The general boson normal ordering problem, Phys. Lett. A 309 (2003) 198-205.
• M. Schork, On the combinatorics of normal ordering bosonic operators and deforming it, J. Phys. A 36 (2003) 4651-4665.

Crossrefs

Cf. A091028 (alternating row sums of A090440).

Programs

Mathematica

ff[n_, k_] = Pochhammer[n - k + 1, k]; a[1, 3] = 1; a[n_, k_] := a[n, k] = Sum[Binomial[3, p]*ff[(n - 1 - p + k), 3 - p]*a[n - 1, k - p], {p, 0, 3} ]; a[n_ /; n < 2, ] = 0; Table[Sum[a[n, k] , {k, 3, 3 n}], {n, 1, 9}] (* _Jean-François Alcover, Sep 01 2011 *)

Formula

In Maple notation, a(n) = (1/12)*n!*(n+1)!*(n+2)!*hypergeom([n+1, n+2, n+3], [2, 3, 4], 1)/exp(1).

a(n) = Sum_{k=3..3*n} A090440(n, k) = (Sum_{k>=3} (1/k!)*Product_{j=1..n} fallfac(k+(j-1)*(4-3), 3))/exp(1), n>=1. From eq.(9) of the Blasiak et al. reference with r=4, s=3. fallfac(n, m) := A008279(n, m) (falling factorials triangle). a(0) := 1 may be added.

Extensions

Edited by Wolfdieter Lang, Dec 23 2003