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A071379
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a(n)=sum(((k+4)!/k!)^(n-1)/k!,k=0..infinity)/exp(1),n=1,2... . This is a Dobinski-type summation formula.
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7
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1, 209, 163121, 326922081, 1346634725665, 9939316337679281, 119802044788535500753, 2205421644124274191535553, 58945667435045762187763602753, 2198513228897522394476415669503377
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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COMMENTS
| a(n) quickly become gigantic: a(15)= 9142140479823239889945170786704021785456107245847570873873. a(n) appears in the process of ordering the n-th power of a product of fourth power of boson creation and fourth power of boson annihilation operators.
Contribution from Peter Luschny, Mar 27 2011: (Start) Let B_{m}(x) = sum_{j>=0}(exp(j!/(j-m)!*x-1)/j!) then a(n) = n! [x^n] taylor(B_{4}(x)), where [x^n] denotes the coefficient of x^k in the Taylor series for B_{4}(x).
a(n) is row 4 of the square array representation of A090210. (End)
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REFERENCES
| P. Blasiak, K. A. Penson and A. I. Solomon, The general boson normal ordering problem, Phys. Lett. A 309 (2003) 198-205.
M. Schork, On the combinatorics of normal ordering bosonic operators and deforming it, J. Phys. A 36 (2003) 4651-4665.
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LINKS
| P. Blasiak, K. A. Penson and A. I. Solomon, The general boson normal ordering problem.
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).
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FORMULA
| sum((fallfac(k, 4)^n)/k!, k=4..infinity)/exp(1), n>=1, with fallfac(n, m) := A08279(n, m) (falling factorials). (From eq.(26) with r=4 of the Schork reference.)
E.g.f. with a(0) := 1: (sum((exp(fallfac(k, 4)*x))/k!, k=4..infinity)+8/3)/exp(1). From top of p. 4656 with r=4 of the Schork reference.
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MAPLE
| A071379 := proc(n) local r, s, i;
if n=0 then 1 else r := [seq(5, i=1..n-1)]; s := [seq(1, i=1..n-1)];
exp(-x)*24^(n-1)*hypergeom(r, s, x); round(evalf(subs(x=1, %), 99)) fi end:
seq(A071379(n), n=1..10); # - Peter Luschny, Mar 30, 2011
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CROSSREFS
| Cf. A000110, A020556 and A069223, when k+4 is replaced by k+1, k+2 or k+3 respectively.
Cf. A090210.
Sequence in context: A029554 A203458 A003779 * A125549 A104876 A050516
Adjacent sequences: A071376 A071377 A071378 * A071380 A071381 A071382
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KEYWORD
| nonn
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AUTHOR
| Karol A. Penson (penson(AT)lptl.jussieu.fr), May 22 2002
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