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%I #44 Sep 02 2018 08:22:21
%S 1,360,259200,1959552000,2821754880000,5079158784000000,
%T 76796880814080000000,304115648023756800000000,
%U 125121866615488512000000000,258236518070374430146560000000000,929651465053347948527616000000000000,334674527419205261469941760000000000000,920050700832433373350094438400000000000000
%N Denominator of the coefficient of z^(-n) in the Stirling-like asymptotic expansion of Product_{z=1..n} z^(z^2).
%C 1^(1^2)*2^(2^2)*...*n^(n^2) ~ A_2*n^(n^3/3+n^2/2+n/6)*exp(-n^3/9+n/12)*(Sum_{k>=0} b(k)/n^k)^n, where A_2 is the second Bendersky constant.
%C a(n) is the denominator of b(n).
%H Seiichi Manyama, <a href="/A317796/b317796.txt">Table of n, a(n) for n = 0..206</a>
%H Weiping Wang, <a href="https://www.researchgate.net/publication/318153972_Some_asymptotic_expansions_on_hyperfactorial_functions_and_generalized_Glaisher-Kinkelin_constants">Some asymptotic expansions on hyperfactorial functions and generalized Glaisher-Kinkelin constants</a>, ResearchGate, 2017.
%F Let B_n be the Bernoulli number, and define the sequence {c_n} by the recurrence
%F c_0 = 1, c_n = (2/n) * Sum_{k=0..n-1} B_{n-k+3}*c_k/((n-j+1)*(n-k+2)*(n-k+3)) for n > 0.
%F a(n) is the denominator of c_n.
%e 1^(1^2)*2^(2^2)*...*n^(n^2) ~ A_2*n^(n^3/3+n^2/2+n/6)*exp(-n^3/9+n/12)*(1 - 1/(360*n) + 1/(259200*n^2) + 259193/(1959552000*n^3) - 1036793/(2821754880000*n^4) - 201551328007/(5079158784000000*n^5) + ... ).
%Y Product_{z=1..n} z^(z^m): A001163/A001164 (m=0), A143475/A143476 (m=1), A317747/A317796 (m=2).
%Y Cf. A051675, A243262 (A_2).
%K nonn,frac
%O 0,2
%A _Seiichi Manyama_, Sep 01 2018
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