A318711 Denominator 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, 720, 7257600, 15676416000, 3476402012160000, 3320318656512000000, 4919915372473221120000000, 4632289550697863577600000000, 507464726196802564122476544000000000, 173072180302909506079665684480000000000, 49554442037561776763544469977956352000000000000

Offset: 0

Author

Seiichi Manyama, Sep 01 2018

Keywords

• nonn
• frac

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 denominator of b(n).

Examples

			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) + ... ).
		

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..149
• 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

Crossrefs

Cf. A143476, A318710.

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 denominator of c_n.