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A182924 Generalized vertical Bell numbers of order 4. 5

%I #24 Oct 24 2018 08:10:06

%S 1,52,43833,149670844,1346634725665,25571928251231076,

%T 893591647147188285577,52327970757667659912764908,

%U 4796836032234830356783078467969,653510798275634770675047022800897940,127014654376520087360456517007106313763801

%N Generalized vertical Bell numbers of order 4.

%C 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 Apparently a(n) = A157280(n+1) for 0 <= n <= 8. - _Georg Fischer_, Oct 24 2018

%H G. C. Greubel, <a href="/A182924/b182924.txt">Table of n, a(n) for n = 0..129</a>

%H 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 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 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 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 := 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;

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

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

%K nonn

%O 0,2

%A _Peter Luschny_, Mar 28 2011

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Last modified March 28 08:02 EDT 2024. Contains 371236 sequences. (Running on oeis4.)