OFFSET
0,3
COMMENTS
G(z+1) ~ A^(-1)*z^(-z^2/2-z/2-1/12)*exp(z^2/4)*(Gamma(z+1))^z*(Sum_{n>=0} b(n)/z^(2*n)), where A is the Glaisher-Kinkelin constant and Gamma is the gamma function.
a(n) is the numerator of b(n).
LINKS
Seiichi Manyama, Table of n, a(n) for n = 0..114
Chao-Ping Chen, Asymptotic expansions for Barnes G-function, Journal of Number Theory 135 (2014) 36-42.
Eric Weisstein's World of Mathematics, Barnes G-Function
FORMULA
Let B_n be the Bernoulli number, and define the sequence {c_n} by the recurrence
c_0 = 1, c_n = (1/(2*n)) * Sum_{k=0..n-1} B_{2*n-2*k+2}*c_k/((2*n-2*k+1)*(2*n-2*k+2)) for n > 0.
a(n) is the numerator of c_n.
EXAMPLE
G(z+1) ~ A^(-1)*z^(-z^2/2-z/2-1/12)*exp(z^2/4)*(Gamma(z+1))^z*(1 - 1/(720*z^2) + 1447/(7257600*z^4) - 1559527/(15676416000*z^6) + 366331136219/(3476402012160000*z^8) - 637231027521743/(3320318656512000000*z^10) + ... ).
CROSSREFS
KEYWORD
sign,frac
AUTHOR
Seiichi Manyama, Sep 01 2018
STATUS
approved