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%I #21 Apr 02 2016 07:04:45
%S 1,73,16333,8030353,7209986401,10541813012041,23227377813664333,
%T 72925401604382826913,312727862321385812968033
%N Generalized Bell numbers B_{4,3}.
%H G. C. Greubel, <a href="/A070531/b070531.txt">Table of n, a(n) for n = 1..200</a>
%H P. Blasiak, K. A. Penson and A. I. Solomon, <a href="http://www.arXiv.org/abs/quant-ph/0402027">The general boson normal ordering problem</a>, arXiv:quant-ph/0402027, 2004.
%H P. Blasiak, K. A. Penson and A. I. Solomon, <a href="http://dx.doi.org/10.1016/S0375-9601(03)00194-4">The general boson normal ordering problem</a>, Phys. Lett. A 309 (2003) 198-205.M. Schork, <a href="http://dx.doi.org/10.1088/0305-4470/36/16/314">On the combinatorics of normal ordering bosonic operators and deforming it</a>, J. Phys. A 36 (2003) 4651-4665.
%F 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)).
%F 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.
%t 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 *)
%Y Cf. A091028 (alternating row sums of A090440).
%K nonn
%O 1,2
%A _Karol A. Penson_, May 02 2002
%E Edited by _Wolfdieter Lang_, Dec 23 2003
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