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A070531 Generalized Bell numbers B_{4,3}. 3
1, 73, 16333, 8030353, 7209986401, 10541813012041, 23227377813664333, 72925401604382826913, 312727862321385812968033 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

LINKS

G. C. Greubel, Table of n, a(n) for n = 1..200

P. Blasiak, K. A. Penson and A. I. Solomon, The general boson normal ordering problem, arXiv:quant-ph/0402027, 2004.

P. Blasiak, K. 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.

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(A090440(n, k), k=3..3*n)= sum((1/k!)*product(fallfac(k+(j-1)*(4-3), 3), j=1..n), k=3..infinity)/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.

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-Fran├žois Alcover, Sep 01 2011 *)

CROSSREFS

Cf. A091028 (alternating row sums of A090440).

Sequence in context: A174747 A210382 A091757 * A274591 A232293 A232366

Adjacent sequences:  A070528 A070529 A070530 * A070532 A070533 A070534

KEYWORD

nonn

AUTHOR

Karol A. Penson, May 02 2002

EXTENSIONS

Edited by Wolfdieter Lang, Dec 23 2003

STATUS

approved

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Last modified December 3 21:14 EST 2016. Contains 278745 sequences.