A318710 - OEIS

%I #25 Sep 01 2018 21:38:35

%S 1,-1,1447,-1559527,366331136219,-637231027521743,

%T 2629597771763437160249,-9781318441276304057417323,

%U 5699253125605574587097648227233017,-13391188869589008440145241321782451523,33214021675956829606886933935672301973543264421

%N 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.

%C 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.

%C a(n) is the numerator of b(n).

%H Seiichi Manyama, <a href="/A318710/b318710.txt">Table of n, a(n) for n = 0..114</a>

%H Chao-Ping Chen, <a href="https://doi.org/10.1016/j.jnt.2013.08.007">Asymptotic expansions for Barnes G-function</a>, Journal of Number Theory 135 (2014) 36-42.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/BarnesG-Function.html">Barnes G-Function</a>

%F Let B_n be the Bernoulli number, and define the sequence {c_n} by the recurrence

%F 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.

%F a(n) is the numerator of c_n.

%e 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) + ... ).

%Y Cf. A143475, A318711.

%K sign,frac

%O 0,3

%A _Seiichi Manyama_, Sep 01 2018