OFFSET
0,3
COMMENTS
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_{5}(x)), where [x^n] denotes the coefficient of x^k in the Taylor series for B_{5}(x).
a(n) is row 5 of the square array representation of A090210. (End)
REFERENCES
M. Schork, On the combinatorics of normal ordering bosonic operators and deforming it, J. Phys. A 36 (2003) 4651-4665.
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..115
P. Blasiak, K. A. Penson and A. I. Solomon, The general boson normal ordering problem., arXiv:quant-ph/0402027, Phys. Lett. A 309 (3-4) (2003) 198-205
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) = Sum_{k=5..5*n} A090216(n, k), n>=1. a(0) := 1.
a(n) = Sum_{k >=5} (fallfac(k, 5)^n)/k!/exp(1), n>=1, a(0) := 1. From eq.(26) with r=5 of the Schork reference.
E.g.f. with a(0) := 1: (sum((exp(fallfac(k, 5)*x))/k!, k=5..infinity)+ A000522(4)/4!)/exp(1). From the top of p. 4656 with r=5 of the Schork reference.
MAPLE
A071379 := proc(n) local r, s, i;
if n=0 then 1 else r := [seq(6, i=1..n-1)]; s := [seq(1, i=1..n-1)];
exp(-x)*5!^(n-1)*hypergeom(r, s, x); round(evalf(subs(x=1, %), 99)) fi end:
seq(A071379(n), n=0..8); # Peter Luschny, Mar 30, 2011
MATHEMATICA
fallfac[n_, k_] := Pochhammer[n-k+1, k]; a[n_, k_] := (((-1)^k)/k!)*Sum[((-1)^p)*Binomial[k, p]*fallfac[p, 5]^n, {p, 5, k}]; a[0] = 1; a[n_] := Sum[a[n, k], {k, 5, 5*n}]; Table[a[n], {n, 0, 8}] (* Jean-François Alcover, Mar 05 2014 *)
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Wolfdieter Lang, Dec 01 2003
EXTENSIONS
If it is proved that A283154 and A090209 are the same, then the entries should be merged and A283154 recycled. - N. J. A. Sloane, Mar 06 2017
STATUS
approved