cp's OEIS Frontend

A182924 Generalized vertical Bell numbers of order 4.

%I A182924 #28 Apr 23 2024 11:25:08
%S A182924 1,52,43833,149670844,1346634725665,25571928251231076,
%T A182924 893591647147188285577,52327970757667659912764908,
%U A182924 4796836032234830356783078467969,653510798275634770675047022800897940,127014654376520087360456517007106313763801
%N A182924 Generalized vertical Bell numbers of order 4.
%C A182924 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.
%C A182924 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).
%H A182924 G. C. Greubel, <a href="/A182924/b182924.txt">Table of n, a(n) for n = 0..129</a>
%H A182924 P. Blasiak and P. Flajolet, <a href="http://arxiv.org/abs/1010.0354">Combinatorial models of creation-annihilation</a>, arXiv:1010.0354 [math.CO], 2010-2011.
%H A182924 K. A. Penson, P. Blasiak, A. Horzela, A. I. Solomon and G. H. E. Duchamp, <a href="http://arxiv.org/abs/0904.0369">Laguerre-type derivatives: Dobinski relations and combinatorial identities</a>, J. Math. Phys. 50, 083512 (2009).
%F A182924 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.
%F A182924 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).
%p A182924 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;
%p A182924 seq(A182924(i),i=0..10);
%t A182924 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 *)
%Y A182924 Cf. A090210, A002720, A069948,A157280, A182925.
%K A182924 nonn
%O A182924 0,2
%A A182924 _Peter Luschny_, Mar 28 2011