A260448 Denominators in the asymptotic expansion of the Barnes G-function.

1, 12, 1440, 51840, 87091200, 1045094400, 376233984000, 902961561600, 166867296583680000, 18021668031037440000, 140569010642092032000000, 1686828127705104384000000, 8501613763633726095360000000, 102019365163604713144320000000, 208119504933753614814412800000000

Offset: 0

Author

Vladimir Reshetnikov, Nov 10 2015

Keywords

• nonn
• frac

Comments

10^(2m)|a(n) where 5m <= n <= 5m+4, m>=0. Example: for m=4, 20<= n <= 24, the values of a(20) to a(24) are divisible by 10^(10). - G. C. Greubel, Dec 15 2015

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.

Crossrefs

Cf. A260447 (numerators), A000178, A001163, A001164.

Programs

Mathematica

Denominator[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 + ...).