OFFSET
0,10
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).
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
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.
EXAMPLE
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.
CROSSREFS
KEYWORD
sign,frac
AUTHOR
Seiichi Manyama, Sep 01 2018
STATUS
approved