OFFSET
0,3
REFERENCES
G. Herzberg, Molecular Spectra and Molecular Structure II: Infrared and Raman Spectra of Polyatomic Molecules, D. Van Nostrand, 1945. see page 505
D. A. McQuarrie, Statistical Mechanics, University Science Books, 2000, see page 100 equ. (6-35)
G. H. Wannier, Statistical Physics, Dover Publications, 1987. see page 216 equ. (11.21)
FORMULA
Sum_{k>=0} (2*k + 1) * exp(-x*(k^2 + k)) ~ (1/x) * Sum_{k>=0} a(k) * (2*x)^k / (2*k + 1)!.
a(n) ~ 2^(n + 7/2) * n^(3*n + 3/2) / (exp(3*n) * Pi^(2*n - 1/2)). - Vaclav Kotesovec, Jun 08 2019
EXAMPLE
1 + 3*exp(-2*x) + 5*exp(-6*x) + 7*exp(-10*x) + ... ~ 1/x + 1/3 + (1/15)*x + (4/315)*x^2 + ...
MATHEMATICA
a[ n_] := If[ n < 0, 0, Sum[ BernoulliB[n + j] / (j! (n - j)!), {j, 0, n }] (2 n + 1)! / (-2)^n]; (* Michael Somos, Dec 04 2013 *)
PROG
(PARI) {a(n) = if( n<0, 0, sum( j=0, n, bernfrac(n+j) / ((n-j)! * j!)) * (2*n + 1)! / (-2)^n)};
CROSSREFS
KEYWORD
nonn
AUTHOR
Michael Somos, May 15 2005
STATUS
approved