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%I #8 Sep 01 2016 10:22:04
%S 1,43,5083,1160113,432168721,238012552651,181520958432283,
%T 182989529196234433,235492729726705299073,376560458072018837889931,
%U 732162019709408940671604091,1700645336651586566571229542193
%N Generalized Bell numbers B_{6,2}.
%D P. Blasiak, K. A. Penson and A. I. Solomon, The general boson normal ordering problem, Phys. Lett. A 309 (2003) 198-205.
%D M. Schork, On the combinatorics of normal ordering bosonic operators and deforming it, J. Phys. A 36 (2003) 4651-4665.
%F a(n)=sum(A091746(n, k), k=2..2*n)= sum((1/k!)*product(fallfac(k+4*(j-1), 2), j=1..n), k=2..infinity)/exp(1), n>=1. From eq.(9) of the Blasiak et al. reference with r=6, s=2. fallfac(n, m) := A008279(n, m) (falling factorials triangle). a(0) := 1 may be added.
%t a[n_] := Sum[Product[FactorialPower[k+4*(j-1), 2], {j, 1, n}]/k!, {k, 2, Infinity}]/E; Table[an = a[n]; Print["a(", n, ") = ", an]; an, {n, 1, 12}] (* _Jean-François Alcover_, Sep 01 2016 *)
%Y Cf. A072019 ( B_{5, 2}).
%K nonn,easy
%O 1,2
%A _Wolfdieter Lang_, Feb 27 2004