A306651 a(n) = Product_{k=1..n} BarnesG(3*k).
1, 288, 36118462464000, 240498631970530185123135341199360000000000
Offset: 1
Keywords
Links
- Eric Weisstein's World of Mathematics, Barnes G-Function.
- Wikipedia, Barnes G-function
Programs
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Mathematica
Table[Product[BarnesG[3*k], {k, 1, n}], {n, 1, 6}] Round[Table[3^(15*n^2/4 - 7*n/12 - 1/4) * E^(Pi/(18*Sqrt[3]) - PolyGamma[1, 1/3]/(12*Sqrt[3]*Pi) - Zeta[3]/(3*Pi^2) + 1/6 + 3*n*(n + 1)*(2*n + 1)/8 + 3*PolyGamma[-3, n + 1] - (3/2)*Derivative[1, 0][Zeta][-2, n] + (1/6)*Derivative[1, 0][Zeta][-2, 3*n] + (7/2)*Derivative[1, 0][Zeta][-1, n + 1/3] + (5/2)*Derivative[1, 0][Zeta][-1, n + 2/3]) * BarnesG[3*n]^(n + 1) * BarnesG[n + 1/3] * Gamma[n]^(5*n/2 - 13/6) / (BarnesG[4/3] * BarnesG[n]^(5/2) * Gamma[n + 1/3]^(n - 1) * Gamma[3*n]^(3*n*(n + 1)/2 - 2/3) * Glaisher^(3*n + 5) * (2*Pi)^(3*(n + 1)^2/4) * n^(3*n^2/2)), {n, 1, 6}]] (* Vaclav Kotesovec, Mar 04 2019 *)
Formula
a(n) ~ (2*Pi)^(3*n^2/4 + n/4 + 1/6) * 3^(3*n^3/2 + 3*n^2/4 - n/3 - 13/72) * n^(3*n^3/2 + 3*n^2/4 - n/3 - 5/72) / (Gamma(1/3)^(1/3) * A^(n + 1/6) * exp(11*n^3/4 + 9*n^2/8 - 5*n/12 - Zeta(3)/(24*Pi^2) - 1/72)), where A is the Glaisher-Kinkelin constant A074962.
a(n) = Product_{k=1..n} (exp(-8*Zeta'(-1)) * 3^(9*k^2/2 - 3*k + 5/12) * (2*Pi)^(1 - 3*k) * Gamma(k)^2 * Gamma(k + 1/3) * (BarnesG(k) * BarnesG(k + 1/3) * BarnesG(k + 2/3))^3).
a(n) = a(n-1)*A296608(n). - R. J. Mathar, Jul 24 2025
Comments