A255674 Decimal expansion of a constant related to the Barnes G-function.

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, 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, 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

Offset: 1

Author

Vaclav Kotesovec, Jul 10 2015

Keywords

• nonn
• cons

Examples

			1.06988379617813356826829257647028132359737354153723273083785714620398...
		

Links

• Eric Weisstein's World of Mathematics, Barnes G-Function
• Wikipedia, Barnes G-function

Crossrefs

Cf. A055746, A306635.

Programs

Mathematica

(* 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}]
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 *)

Formula

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.

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))).

From Vaclav Kotesovec, Mar 02 2019: (Start)

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.

Equals exp(-1/24 - 7*Zeta(3)/(32*Pi^2) + Zeta'(-1) + log(2)/8 + 3*log(Pi)/16).

(End)