This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A318710 #26 Feb 16 2025 08:33:56
%S A318710 1,-1,1447,-1559527,366331136219,-637231027521743,
%T A318710 2629597771763437160249,-9781318441276304057417323,
%U A318710 5699253125605574587097648227233017,-13391188869589008440145241321782451523,33214021675956829606886933935672301973543264421
%N 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.
%C A318710 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 A318710 a(n) is the numerator of b(n).
%H A318710 Seiichi Manyama, <a href="/A318710/b318710.txt">Table of n, a(n) for n = 0..114</a>
%H A318710 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 A318710 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/BarnesG-Function.html">Barnes G-Function</a>
%F A318710 Let B_n be the Bernoulli number, and define the sequence {c_n} by the recurrence
%F A318710 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 A318710 a(n) is the numerator of c_n.
%e A318710 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 A318710 Cf. A143475, A318711.
%K A318710 sign,frac
%O A318710 0,3
%A A318710 _Seiichi Manyama_, Sep 01 2018