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A318713 Numerator of the coefficient of z^(-2*n) in the Stirling-like asymptotic expansion of Product_{z=1..n} z^(z^3).

%I A318713 #25 Sep 03 2018 01:53:46
%S A318713 1,-1,1513,-127057907,7078687551763,-1626209947417109183,
%T A318713 25620826938516570309695021,-67861652779316417663427293866727,
%U A318713 11129902336987204608540488473560076627,-2992048697379116617363098289271338606184087563,593799837691907572156765292649932318031816367209421
%N A318713 Numerator of the coefficient of z^(-2*n) in the Stirling-like asymptotic expansion of Product_{z=1..n} z^(z^3).
%C A318713 1^(1^3)*2^(2^3)*...*n^(n^3) ~ A_3*n^(n^4/4+n^3/2+n^2/4-1/120)*exp(-n^4/16+n^/12)*(Sum_{k>=0} b(k)/n^k)^n, where A_3 is the third Bendersky constant.
%C A318713 a(n) is the numerator of b(n).
%H A318713 Seiichi Manyama, <a href="/A318713/b318713.txt">Table of n, a(n) for n = 0..106</a>
%H A318713 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 A318713 Let B_n be the Bernoulli number, and define the sequence {c_n} by the recurrence
%F A318713 c_0 = 1, c_n = (-3/n) * Sum_{k=0..n-1} B_{2*n-2*k+4}*c_k/((2*n-2*k+1)*(2*n-2*k+2)*(2*n-2*k+3)*(2*n-2*k+4)) for n > 0.
%F A318713 a(n) is the numerator of c_n.
%e A318713 1^(1^3)*2^(2^3)*...*n^(n^3) ~ A_3*n^(n^4/4+n^3/2+n^2/4-1/120)*exp(-n^4/16+n^/12)*(1 - 1/(5040*n^2) + 1513/(50803200*n^4) - 127057907/(8449588224000*n^6) + 7078687551763/(442893616349184000*n^8) - 1626209947417109183/(55804595659997184000000*n^10) + ... ).
%Y A318713 Product_{z=1..n} z^(z^m): A001163/A001164 (m=0), A143475/A143476 (m=1), A317747/A317796 (m=2), A318713/A318714 (m=3).
%Y A318713 Cf. A243263 (A_3).
%K A318713 sign,frac
%O A318713 0,3
%A A318713 _Seiichi Manyama_, Sep 01 2018