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%I #24 Apr 19 2023 19:00:33
%S 1,1,-1433,1550887,-365236274341,31170363588856607,
%T -2626723351027654662151,127061942835077684151157039,
%U -5696145248370283185291966600124423,254326794362835881966596504823903633657,-33203124408022060010631772664020406983485604379
%N Numerator of the coefficient of z^(2n) in the Stirling-like asymptotic expansion of the hyperfactorial function A002109.
%D Mohammad K. Azarian, On the Hyperfactorial Function, Hypertriangular Function, and the Discriminants of Certain Polynomials, International Journal of Pure and Applied Mathematics, Vol. 36, No. 2, 2007, pp. 251-257. Mathematical Reviews, MR2312537. Zentralblatt MATH, Zbl 1133.11012.
%H Seiichi Manyama, <a href="/A143475/b143475.txt">Table of n, a(n) for n = 0..113</a>
%H Jean-Christophe Pain, <a href="https://arxiv.org/abs/2304.07629">Series representations for the logarithm of the Glaisher-Kinkelin constant</a>, arXiv:2304.07629 [math.NT], 2023.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Hyperfactorial.html">Hyperfactorial</a>
%F From _Seiichi Manyama_, Aug 31 2018: (Start)
%F Let B_n be the Bernoulli number, and define the sequence {c_n} by the recurrence
%F c_0 = 1, c_n = (-1/(2*n)) * Sum_{k=0..n-1} B_{2*n-2*k+2}*c_k/((2*n-2*k+1)*(2*n-2*k+2)) for n > 0.
%F a(n) is the numerator of c_n. (End)
%e (Glaisher*(1 - 1433/(7257600*z^4) + 1/(720*z^2))*z^(1/12 + (z*(1 + z))/2))/e^(z^2/4).
%e From _Seiichi Manyama_, Aug 31 2018: (Start)
%e c_1 = -1/2 * (B_4*c_0/(3*4)) = 1/720, so a(1) = 1.
%e c_2 = -1/4 * (B_6*c_0/(5*6) + B_4*c_1/(3*4)) = -1433/7257600, so a(2) = -1433. (End)
%Y Cf. A002109, A143476.
%K sign,frac
%O 0,3
%A _Eric W. Weisstein_, Aug 19 2008
%E More terms from _Seiichi Manyama_, Aug 31 2018
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