A260447 Numerators in the asymptotic expansion of the Barnes G-function.
1, -1, -1, 157, 65911, -918227, -234932171, 592642957, 149463069056137, -16648792617135289, -286497627060094895989, 3538603031642540133299, 57674522110226646157873673, -713986824035720029666660757, -6478234620955890989297122598683
Offset: 0
Links
- G. C. Greubel and D. Turner, Table of n, a(n) for n = 0..175
- Eric Weisstein's World of Mathematics, Barnes G-Function.
- Wikipedia, Barnes G-function.
Programs
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Mathematica
Numerator[Exp[Series[LogBarnesG[x] - 1/12 - x + 3 x^2/4 + Log[Glaisher] + Log[2 Pi]/2 - x Log[2 Pi]/2 - 5 Log[x]/12 + x Log[x] - x^2 Log[x]/2, {x, Infinity, 20}]][[3]]]
Formula
G(x) ~ exp^(-3*x^2/4 + x + zeta'(-1)) * x^(x^2/2 - x + 5/12) * (2*Pi)^((x-1)/2) * (1 + (-1/12)/x + (-1/1440)/x^2 + (157/51840)/x^3 + (65911/87091200)/x^4 + ...).