

A182925


Generalized vertical Bell numbers of order 3.


4



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

0,2


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

K. A. Penson, P. Blasiak, A. Horzela, A. I. Solomon and G. H. E. Duchamp,


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!/(jn)!*x1)/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).


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}] (* JeanFrançois Alcover, Nov 22 2012, after Maple *)


CROSSREFS



KEYWORD

nonn


AUTHOR



STATUS

approved



