OFFSET
1,2
LINKS
P. Blasiak, K. A. Penson, and A. I. Solomon, The general boson normal ordering problem, Phys. Lett. A 309 (2003) 198-205.
P. Blasiak, K. A. Penson, and A. I. Solomon, The general boson normal ordering problem, arXiv:quant-ph/0402027, 2004.
Wolfdieter Lang, First 10 rows.
M. Schork, On the combinatorics of normal ordering bosonic operators and deforming it, J. Phys. A 36 (2003) 4651-4665.
FORMULA
a(n, m) = m^(2*(n-m))*Pochhammer(1/m, n-m)*Pochhammer(2/m, n-m)/2 if n-1 >= m >= 1; a(n, 0) = 2^(n-1); otherwise 0.
E.g.f. for m = 1, 2, ... column (without leading zeros and offset n=1): (hypergeom([1/m, 2/m], [], (m^2)*x)-1)/2.
G.f. for m=1 column: x/(1-2*x); e.g.f.: (exp(2*x)-1)/2.
a(n, m) = (1/2)*Product_{j=0..n-m-1} (m*j+2)*(m*j+1), n >= m+1 >= 1, otherwise 0. From eq. 12 of the Blasiak et al. reference with r=m+2, s=2, k=2.
CROSSREFS
KEYWORD
AUTHOR
Wolfdieter Lang, Feb 13 2004
STATUS
approved