cp's OEIS Frontend

A182925 Generalized vertical Bell numbers of order 3.

%I A182925 #17 May 17 2018 21:49:55
%S A182925 1,15,1657,513559,326922081,363303011071,637056434385865,
%T A182925 1644720885001919607,5943555582476814384769,
%U A182925 28924444943026683877502191,183866199607767992029159792281,1489437787210535537087417039489815
%N A182925 Generalized vertical Bell numbers of order 3.
%C A182925 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 A182925 G. C. Greubel, <a href="/A182925/b182925.txt">Table of n, a(n) for n = 0..168</a>
%H A182925 P. Blasiak and P. Flajolet, <a href="http://arxiv.org/abs/1010.0354">Combinatorial models of creation-annihilation</a>, (2010).
%H A182925 K. A. Penson, P. Blasiak, A. Horzela, A. I. Solomon and G. H. E. Duchamp,
%H A182925 <a href="http://arxiv.org/abs/0904.0369">Laguerre-type derivatives: Dobinski relations and combinatorial identities</a>, J. Math. Phys. 50, 083512 (2009).
%F A182925 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 A182925 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 A182925 := proc(n) exp(-x)*GAMMA(n+1)^3*hypergeom([n+1,n+1,n+1],[1,1,1],x);
%p A182925 round(evalf(subs(x=1,%),64)) end; seq(A182925(i),i=0..11);
%t A182925 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 A182925 Cf. A090210, A002720, A069948, A182924.
%K A182925 nonn
%O A182925 0,2
%A A182925 _Peter Luschny_, Mar 28 2011