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%I #15 Mar 02 2019 11:09:52
%S 1,0,6,9,8,8,3,7,9,6,1,7,8,1,3,3,5,6,8,2,6,8,2,9,2,5,7,6,4,7,0,2,8,1,
%T 3,2,3,5,9,7,3,7,3,5,4,1,5,3,7,2,3,2,7,3,0,8,3,7,8,5,7,1,4,6,2,0,3,9,
%U 8,6,3,0,9,0,7,2,2,3,1,3,3,7,7,2,7,0,8,5,9,8,9,9,3,0,5,9,6,8,0,3,5,7,0,5,4
%N Decimal expansion of a constant related to the Barnes G-function.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/BarnesG-Function.html">Barnes G-Function</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Barnes_G-function">Barnes G-function</a>
%F Equals limit n->infinity (Product_{j = 1..n} BarnesG(j + 1/2) / BarnesG(j)) / (A^(1/2) * n^(n^2/4 - n/8 - 1/24) * (2*Pi)^(n/4 - 3/16) / exp(n*(3*n-1)/8)), where A = A074962 = 1.2824271291... is the Glaisher-Kinkelin constant.
%F Equals limit n->infinity A055746(n) / (2^(n^3/3 + n^2 - n/8 - 71/48) * exp(9*n^2/8 + 5*n/2 - 7/24) * A^(3*n/2 + 4) / (n^(3*n^2/4 + 21*n/8 + 9/4) * Pi^(n^2/4 + 5*n/4 + 27/16))).
%F From _Vaclav Kotesovec_, Mar 02 2019: (Start)
%F Equals 2^(1/8) * Pi^(3/16) * exp(1/24 - 7*Zeta(3)/(32*Pi^2)) / A, where A is the Glaisher-Kinkelin constant A074962.
%F Equals exp(-1/24 - 7*Zeta(3)/(32*Pi^2) + Zeta'(-1) + log(2)/8 + 3*log(Pi)/16).
%F (End)
%e 1.06988379617813356826829257647028132359737354153723273083785714620398...
%t (* The iteration cycle: *) $MaxExtraPrecision = 1000; funs[n_]:=Product[BarnesG[j+1/2] / BarnesG[j], {j, 1, n}] / (Glaisher^(1/2) * n^(n^2/4 - n/8 - 1/24) * (2*Pi)^(n/4 - 3/16) / E^(n*(3*n-1)/8)); Do[Print[N[Sum[(-1)^(m + j)*funs[j*Floor[200/m]]*(j^(m - 1)/(j - 1)!/(m - j)!), {j, 1, m}], 120]], {m, 10, 150, 10}]
%t RealDigits[2^(1/8) * Pi^(3/16) * E^(1/24 - 7*Zeta[3]/(32*Pi^2)) / Glaisher, 10, 120][[1]] (* _Vaclav Kotesovec_, Mar 02 2019 *)
%Y Cf. A055746, A306635.
%K nonn,cons
%O 1,3
%A _Vaclav Kotesovec_, Jul 10 2015
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