OFFSET
1,2
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.
LINKS
P. Blasiak, K. A. Penson and A. I. Solomon, The general boson normal ordering problem.
FORMULA
a(n) := sum( A078741(n, k)*(-1)^(k+1), k=3..3*n), n>=1. a(0) := -1 may be added.
a(n) = -sum(((-1)^k)*(fallfac(k, 3)^n)/k!, k=3..infinity)*exp(1), with fallfac(k, 3)=A008279(k, 3)=k*(k-1)*(k-2) and n>=1. This produces also a(0)=-1.
E.g.f. if a(0)=-1 is added: -exp(1)*(sum(((-1)^k)*exp(fallfac(k, 3)*x)/k!, k=3..infinity)+1/2). Similar to derivation on top of p. 4656 of the Schork reference.
MATHEMATICA
a[n_] := -Sum[(-1)^k FactorialPower[k, 3]^n/k!, {k, 2, Infinity}]*E; Array[a, 14] (* Jean-François Alcover, Sep 01 2016 *)
CROSSREFS
KEYWORD
sign,easy
AUTHOR
Wolfdieter Lang, Dec 01 2003
STATUS
approved