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A182924 Generalized vertical Bell numbers of order 4. 5
1, 52, 43833, 149670844, 1346634725665, 25571928251231076, 893591647147188285577, 52327970757667659912764908, 4796836032234830356783078467969, 653510798275634770675047022800897940, 127014654376520087360456517007106313763801 (list; graph; refs; listen; history; text; internal format)
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 5 in this representation. The order is the parameter M in Penson et al., p. 6, eq. 29.

Apparently a(n) = A157280(n+1) for 0 <= n <= 8. - Georg Fischer, Oct 24 2018

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..129

P. Blasiak and P. Flajolet, Combinatorial models of creation-annihilation, arXiv:1010.0354 [math.CO], 2010-2011.

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).

FORMULA

a(n) = exp(-1)*Gamma(n+1)^4*[4F4]([n+1,n+1,n+1,n+1], [1,1,1,1] | 1); here [4F4] is the generalized hypergeometric function of type 4F4.

Let B_{n}(x) = sum_{j>=0}(exp(j!/(j-n)!*x-1)/j!) then a(n) = 5! [x^5] taylor(B_{n}(x)), where [x^5] denotes the coefficient of x^5 in the Taylor series for B_{n}(x).

MAPLE

A182924 := proc(n) exp(-x)*GAMMA(n+1)^4*hypergeom([n+1, n+1, n+1, n+1], [1, 1, 1, 1], x); round(evalf(subs(x=1, %), 99)) end;

seq(A182924(i), i=0..10);

MATHEMATICA

fallfac[n_, k_] := Pochhammer[n-k+1, k]; f[m_][n_, k_] := (-1)^k/k!* Sum[(-1)^p*Binomial[k, p]*fallfac[p, m]^n, {p, m, k}]; a[n_] := Sum[f[n][5, k], {k, n, 5*n}]; Table[a[n], {n, 0, 10}] (* Jean-Fran├žois Alcover, Sep 05 2012 *)

CROSSREFS

Cf. A090210, A002720, A069948,A157280, A182925.

Sequence in context: A068255 A230532 A157280 * A208785 A206388 A263223

Adjacent sequences:  A182921 A182922 A182923 * A182925 A182926 A182927

KEYWORD

nonn

AUTHOR

Peter Luschny, Mar 28 2011

STATUS

approved

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Last modified February 24 07:19 EST 2020. Contains 332199 sequences. (Running on oeis4.)