A182925 Generalized vertical Bell numbers of order 3.

1, 15, 1657, 513559, 326922081, 363303011071, 637056434385865, 1644720885001919607, 5943555582476814384769, 28924444943026683877502191, 183866199607767992029159792281, 1489437787210535537087417039489815

Offset: 0

Author

Peter Luschny, Mar 28 2011

Keywords

• nonn

Comments

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.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..168
• P. Blasiak and P. Flajolet, Combinatorial models of creation-annihilation, (2010).
• K. A. Penson, P. Blasiak, A. Horzela, A. I. Solomon and G. H. E. Duchamp,
• Laguerre-type derivatives: Dobinski relations and combinatorial identities, J. Math. Phys. 50, 083512 (2009).

Crossrefs

Cf. A090210, A002720, A069948, A182924.

Programs

Maple

A182925 := proc(n) exp(-x)*GAMMA(n+1)^3*hypergeom([n+1,n+1,n+1],[1,1,1],x);
round(evalf(subs(x=1,%),64)) end; seq(A182925(i),i=0..11);

Mathematica

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 *)

Formula

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.

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).