A317615 Numerator of the coefficient of z^(-n) of asymptotic expansions related to hyperfactorial function H(z).

1, 0, 0, 1, 0, -1, 1, 1, -1, -2591989, 1, 143285617, -576460261, -56447987, 3300470359, 3143536325702272141, -1786064113, -75094364111707432631, 2817261842693900500163, 669559278432109042489, -328057861344412119208697, -10210033960347221028684507558893, 4858439165868890899345523

Offset: 0

Author

Seiichi Manyama, Sep 01 2018

Keywords

• sign
• frac

Comments

1^1*2^2*...*n^n ~ A*n^(n^2/2+n/2+1/12)*exp(-n^2/4)*(Sum_{k>=0} b(k)/n^k)^n, where A is the Glaisher-Kinkelin constant.

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

Examples

			1^1*2^2*...*n^n ~ A*n^(n^2/2+n/2+1/12)*exp(-n^2/4)*(1 + 1/(720*n^3) - 1/(5040*n^5) + 1/(1036800*n^6) + 1/(10080*n^7) - 1/(3628800*n^8) - 2591989/(24634368000*n^9) + ... )^n.
		

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..266
• Chao-Ping Chen, Asymptotic expansions for Barnes G-function, Journal of Number Theory 135 (2014) 36-42.
• Eric Weisstein's World of Mathematics, Hyperfactorial

Crossrefs

Cf. A143475, A317660.

Formula

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

c_0 = 1, c_n = (-1/n) * Sum_{k=0..n-2} B_{n-k+1}*c_k/((n-k-1)*(n-k+1)) for n > 0.

a(n) is the numerator of c_n.