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A070531
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Generalized Bell numbers B_{4,3}.
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2
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OFFSET
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1,2
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REFERENCES
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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.
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LINKS
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Table of n, a(n) for n=1..9.
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FORMULA
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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.
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MATHEMATICA
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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 *)
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CROSSREFS
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Cf. A091028 (alternating row sums of A090440).
Sequence in context: A174747 A210382 A091757 * A232293 A232366 A183490
Adjacent sequences: A070528 A070529 A070530 * A070532 A070533 A070534
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KEYWORD
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nonn
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AUTHOR
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Karol A. Penson, May 02 2002
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EXTENSIONS
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Edited by Wolfdieter Lang, Dec 23 2003
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STATUS
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approved
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