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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-François 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 June 17 17:41 EDT 2021. Contains 345085 sequences. (Running on oeis4.)