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%I #35 Feb 08 2024 01:56:37
%S 1,0,0,9,6,8,0,3,8,7,2,8,5,8,6,6,1,6,1,1,2,0,0,8,9,1,9,0,4,6,2,6,3,0,
%T 6,9,2,6,0,3,2,7,6,3,4,7,2,1,1,5,2,4,9,1,8,4,6,0,9,2,4,7,2,1,5,6,2,3,
%U 0,1,4,2,5,0,0,3,4,1,0,0,3,2,7,7,0,1,5,0,5,6,5,9,6,5,2,7,6,4,5,5,5,9,4
%N Decimal expansion of the generalized Glaisher-Kinkelin constant A(5).
%C Also known as the 5th Bendersky constant.
%D Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.15 Glaisher-Kinkelin constant, p. 137.
%H G. C. Greubel, <a href="/A243265/b243265.txt">Table of n, a(n) for n = 1..2004</a>
%H Victor S. Adamchik, <a href="https://doi.org/10.1016/S0377-0427(98)00192-7">Polygamma functions of negative order</a>, Journal of Computational and Applied Mathematics, Vol. 100, No. 2 (1998), pp. 191-199.
%H L. Bendersky, <a href="https://doi.org/10.1007/BF02547794">Sur la fonction gamma généralisée</a>, Acta Mathematica , Vol. 61 (1933), pp. 263-322; <a href="https://projecteuclid.org/journals/acta-mathematica/volume-61/issue-none/Sur-la-fonction-gamma-g%C3%A9n%C3%A9ralis%C3%A9e/10.1007/BF02547794.full">alternative link</a>.
%H Robert A. Van Gorder, <a href="https://doi.org/10.1142/S1793042112500297">Glaisher-type products over the primes</a>, International Journal of Number Theory, Vol. 8, No. 2 (2012), pp. 543-550.
%H Eric Weisstein's MathWorld, <a href="http://mathworld.wolfram.com/Glaisher-KinkelinConstant.html">Glaisher-Kinkelin Constant</a>.
%F A(k) = exp(B(k+1)/(k+1)*H(k)-zeta'(-k)), where B(k) is the k-th Bernoulli number and H(k) the k-th harmonic number.
%F A(5) = exp(137/15120-zeta'(-5)).
%F Equals exp(gamma/252 - 15*Zeta'(6)/(4*Pi^6)) * (2*Pi)^(1/252), where gamma is the Euler-Mascheroni constant A001620. - _Vaclav Kotesovec_, Jul 25 2015
%F Equals (2*Pi*exp(gamma) * Product_{p prime} p^(1/(p^6-1)))^c, where gamma is Euler's constant (A001620), and c = Bernoulli(6)/6 = 1/252 (Van Gorder, 2012). - _Amiram Eldar_, Feb 08 2024
%e 1.00968038728586616112008919046263...
%t RealDigits[Exp[137/15120-Zeta'[-5]], 10, 103] // First
%t RealDigits[Exp[N[(BernoulliB[6]/6)*(EulerGamma + Log[2*Pi] - Zeta'[6]/Zeta[6]), 200]]]//First (* _G. C. Greubel_, Dec 31 2015 *)
%Y Cf. A001620, A255344, A259070.
%Y Cf. A019727, A074962, A243262, A243263, A243264, A266553, A266554, A266555, A266556, A266557, A266558, A266559, A260662, A266560, A266562, A266563, A266564, A266565, A266566, A266567.
%K nonn,cons
%O 1,4
%A _Jean-François Alcover_, Jun 02 2014
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