A182924 Generalized vertical Bell numbers of order 4.

1, 52, 43833, 149670844, 1346634725665, 25571928251231076, 893591647147188285577, 52327970757667659912764908, 4796836032234830356783078467969, 653510798275634770675047022800897940, 127014654376520087360456517007106313763801

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 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 (and true considering the hypergeometric comment in A157280, R. J. Mathar, Apr 23 2024).

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).

Crossrefs

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

Programs

Maple

A182924 := proc(n) exp(-x)*GAMMA(n+1)^4*hypergeom([n+1,n+1,n+1,n+1],[1,1,1,1],x): simplify(subs(x=1, %)) 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-François Alcover, Sep 05 2012 *)

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).