login
Denominator of the coefficient of z^(-n) of asymptotic expansions related to hyperfactorial function H(z).
2

%I #51 Sep 01 2018 14:41:52

%S 1,1,1,720,1,5040,1036800,10080,3628800,24634368000,6350400,

%T 747242496000,3476402012160000,105670656000,11298306539520000,

%U 1489290622009344000000,2259661307904000,6688268793387417600000,920024174652492349440000000,8655406673795481600000

%N Denominator 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 denominator of b(n).

%H Seiichi Manyama, <a href="/A317660/b317660.txt">Table of n, a(n) for n = 0..362</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/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 denominator 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. A143476, A317615.

%K nonn,frac

%O 0,4

%A _Seiichi Manyama_, Sep 01 2018