OFFSET
1,2
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 1..240
P. Blasiak, K. A. Penson and A. I. Solomon, The Boson Normal Ordering Problem and Generalized Bell Numbers, arXiv:quant-ph/0212072, 2002.
P. Blasiak, K. A. Penson and A. I. Solomon, The general boson normal ordering problem, arXiv:quant-ph/0402027, 2004.
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.
FORMULA
a(n) = Sum_{k=2..2*n} A078740(n, k) = Sum_{k=1..infinity} (1/k!)*Product_{j=1..n}(fallfac(k+(j-1)*(3-2), 2))/exp(1), n>=1. From eq.(9) of the Blasiak et al. reference with r=3, s=2. fallfac(n, m) := A008279(n, m) (falling factorials triangle). a(0) := 1 may be added.
a(n) = Sum_{k=0..infinity}((n+k)!*(n+k+1)!/(k!*(k+1)!*(k+2)!))/exp(1), n>=1. From eq.(40) of the Blasiak et al. reference. [corrected by Vaclav Kotesovec, Jul 27 2018]
E.g.f. for a(n)/n! with a(0)=(exp(1)-1)/exp(1) added: hypergeom([k+2, k+1], [1], z)/(k+2)!, k=0..infinity)/exp(1)). From eq. (41) of the Blasiak et al. reference.
MATHEMATICA
a[n_] := (n+1)*n!^2*Sum[(-1)^k*HypergeometricPFQ[{2-k, n+1, n+2}, {2, 3}, 1]/(2*(k-2)!), {k, 2, 2n}]; Array[a, 13] (* Jean-François Alcover, Sep 01 2015 *)
Table[Sum[(n + k)!*(n + k + 1)!/(k!*(k + 1)!*(k + 2)!), {k, 0, Infinity}]/E, {n, 1, 20}] (* Vaclav Kotesovec, Jul 27 2018 *)
PROG
(PARI) nmax = 20; p = floor(3*nmax*log(nmax)); default(realprecision, p);
for(n=1, nmax, print1(round(exp(-1)*suminf(k=0, (n+k)!*(n+k+1)!/(k!*(k+1)!*(k+2)!))), ", ")) \\ G. C. Greubel and Vaclav Kotesovec, Jul 28 2018
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
N. J. A. Sloane, Dec 21 2002
EXTENSIONS
Edited by Wolfdieter Lang, Dec 23 2003
STATUS
approved