A318710 Numerator of the coefficient of z^(-2*n) in the Stirling-like asymptotic expansion of G(z+1), where G(z) is Barnes G-function.

1, -1, 1447, -1559527, 366331136219, -637231027521743, 2629597771763437160249, -9781318441276304057417323, 5699253125605574587097648227233017, -13391188869589008440145241321782451523, 33214021675956829606886933935672301973543264421

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

Cf. A143475, A318711.

Sequence in context: A181973 A226097 A023311 * A208486 A242038 A045099

Adjacent sequences: A318707 A318708 A318709 * A318711 A318712 A318713

Keyword

sign,frac

Author

Seiichi Manyama, Sep 01 2018

Status

approved