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%I #17 May 17 2018 21:49:55
%S 1,15,1657,513559,326922081,363303011071,637056434385865,
%T 1644720885001919607,5943555582476814384769,
%U 28924444943026683877502191,183866199607767992029159792281,1489437787210535537087417039489815
%N Generalized vertical Bell numbers of order 3.
%C The name "generalized 'vertical' Bell numbers" is used to distinguish them from the generalized (horizontal) Bell numbers with reference to the square array representation of the generalized Bell numbers as given in A090210. a(n) is column 4 in this representation. The order is the parameter M in Penson et al., p. 6, eq. 29.
%H G. C. Greubel, <a href="/A182925/b182925.txt">Table of n, a(n) for n = 0..168</a>
%H P. Blasiak and P. Flajolet, <a href="http://arxiv.org/abs/1010.0354">Combinatorial models of creation-annihilation</a>, (2010).
%H K. A. Penson, P. Blasiak, A. Horzela, A. I. Solomon and G. H. E. Duchamp,
%H <a href="http://arxiv.org/abs/0904.0369">Laguerre-type derivatives: Dobinski relations and combinatorial identities</a>, J. Math. Phys. 50, 083512 (2009).
%F a(n) = exp(-1)*Gamma(n+1)^3*[3F3]([n+1, n+1, n+1], [1, 1, 1] | 1); here [3F3] is the generalized hypergeometric function of type 3F3.
%F Let B_{n}(x) = Sum_{j>=0}(exp(j!/(j-n)!*x-1)/j!) then a(n) = 4! [x^4] taylor(B_{n}(x)), where [x^4] denotes the coefficient of x^4 in the Taylor series for B_{n}(x).
%p A182925 := proc(n) exp(-x)*GAMMA(n+1)^3*hypergeom([n+1,n+1,n+1],[1,1,1],x);
%p round(evalf(subs(x=1,%),64)) end; seq(A182925(i),i=0..11);
%t u = 1.`64; a[n_] := n!^3*HypergeometricPFQ[{n+u, n+u, n+u}, {u, u, u}, u]/E // Round; Table[a[n], {n, 0, 11}] (* _Jean-François Alcover_, Nov 22 2012, after Maple *)
%Y Cf. A090210, A002720, A069948, A182924.
%K nonn
%O 0,2
%A _Peter Luschny_, Mar 28 2011
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