A317615 - OEIS

%I #41 Feb 16 2025 08:33:56

%S 1,0,0,1,0,-1,1,1,-1,-2591989,1,143285617,-576460261,-56447987,

%T 3300470359,3143536325702272141,-1786064113,-75094364111707432631,

%U 2817261842693900500163,669559278432109042489,-328057861344412119208697,-10210033960347221028684507558893,4858439165868890899345523

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

%C 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.

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

%H Seiichi Manyama, <a href="/A317615/b317615.txt">Table of n, a(n) for n = 0..266</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="https://mathworld.wolfram.com/Hyperfactorial.html">Hyperfactorial</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/n) * Sum_{k=0..n-2} B_{n-k+1}*c_k/((n-k-1)*(n-k+1)) for n > 0.

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

%e 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.

%Y Cf. A143475, A317660.

%K sign,frac,changed

%O 0,10

%A _Seiichi Manyama_, Sep 01 2018