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A069223
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Generalized Bell numbers.
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11
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1, 34, 2971, 513559, 149670844, 66653198353, 42429389528215, 36788942253042556, 41888564490333642283, 60862147523250910055785, 110264570238241604072673394, 244397290937585028603794094349, 652229940568729289038518033117685, 2067551365133160531453420400711013314, 7694635622932764203876848262780670955447
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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COMMENTS
| a(n) occurs in the process of normal ordering of the n-th power of a product of the cubes of the boson creation and boson annihilation operators.
a(11)=110264570238241604072673394 =~ 10^26.
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_{3}(x)), where [x^n] denotes the coefficient of x^n in the Taylor series for B_{3}(x).
a(n) is row 3 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 Boson Normal Ordering Problem and Generalized Bell Numbers
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
| a(n)= sum((((k+3)!)^n)/((k+3)!*(k!)^n), k=0..infinity)/exp(1), n>=1. This is a Dobinski-type summation formula.
a(n)= (sum(((k*(k-1)*(k-2))^n)/k!, k=3..infinity)/exp(1), n>=1. Usually a(0) := 1. (From eq.(26) with r=3 of the Schork reference; rewritten original eq.(25) with r=3 of the Blasiak et al. reference.)
E.g.f. with a(0) := 1: (sum((exp(k*(k-1)*(k-2)*x))/k!, k=3..infinity)+5/2)/exp(1). From top of p. 4656 with r=3 of the Schork reference.
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MAPLE
| A069223 := proc(n) local r, s, i;
if n=0 then 1 else r := [seq(4, i=1..n-1)]; s := [seq(1, i=1..n-1)];
exp(-x)*6^(n-1)*hypergeom(r, s, x); round(evalf(subs(x=1, %), 99)) fi end:
seq(A069223(n), n=1..15); # - Peter Luschny, Mar 30, 2011
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MATHEMATICA
| f[n_] := f[n] = Sum[(k + 3)!^n/((k + 3)!*(k!^n)*E), {k, 0, Infinity}]; Table[ f[n], {n, 1, 9}]
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CROSSREFS
| Cf. A000110 and A020556, if k+3 is replaced by k+1 or k+2, respectively.
Cf. A090210.
Sequence in context: A187591 A160471 A138590 * A129056 A187708 A199837
Adjacent sequences: A069220 A069221 A069222 * A069224 A069225 A069226
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KEYWORD
| nonn,easy
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AUTHOR
| Karol A. Penson (penson(AT)lptl.jussieu.fr), Apr 12 2002
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EXTENSIONS
| Edited by Robert G. Wilson v (rgwv(AT)rgwv.com), Apr 30 2002
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