A324993 -id:A324993 - cp's OEIS frontend

A306651 a(n) = Product_{k=1..n} BarnesG(3*k).

Original entry on oeis.org

1, 288, 36118462464000, 240498631970530185123135341199360000000000

Offset: 1

Vaclav Kotesovec, Mar 03 2019

nonn

Next term is too long to be included.

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

Cf. A055462, A268504, A296608, A306635, A324993.

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 *)

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

A324994 Decimal expansion of zeta'(-1, 2/3) (negated).

Original entry on oeis.org

0, 1, 3, 9, 6, 2, 4, 4, 7, 3, 1, 2, 3, 7, 0, 7, 4, 3, 8, 8, 0, 3, 4, 4, 6, 0, 4, 4, 4, 1, 4, 0, 9, 2, 6, 3, 9, 8, 8, 5, 7, 6, 6, 5, 9, 9, 8, 8, 1, 2, 4, 3, 1, 7, 1, 8, 4, 8, 4, 1, 3, 9, 7, 5, 7, 4, 9, 0, 3, 3, 7, 2, 9, 8, 4, 8, 3, 3, 2, 6, 2, 8, 5, 6, 2, 5, 6, 4, 5, 3, 5, 5, 4, 2, 4, 9, 7, 0, 3, 6, 2, 1, 5, 1, 0, 6

Offset: 0

Vaclav Kotesovec, Mar 23 2019

			-0.01396244731237074388034460444140926398857665998812431718484139757490...

J. Miller and V. Adamchik, Derivatives of the Hurwitz Zeta Function for Rational Arguments, Journal of Computational and Applied Mathematics 100 (1998) 201-206. [contains a large number of typos]
Eric Weisstein's World of Mathematics, Hurwitz Zeta Function, formula 23

Cf. A084448, A240966, A324993.

evalf(Zeta(1,-1,2/3), 120);
evalf(Pi/(18*sqrt(3)) - log(3)/72 - Psi(1, 1/3) / (12*sqrt(3)*Pi) - Zeta(1,-1)/3, 120);

RealDigits[Derivative[1, 0][Zeta][-1, 2/3], 10, 120][[1]]
N[With[{k=1}, Sqrt[3] * (9^k - 1) * BernoulliB[2*k] * Pi / (9^k * 8*k) - 3*BernoulliB[2*k] * Log[3] / 9^k / 4 / k + (-1)^k * PolyGamma[2*k-1,1/3] / 2 / Sqrt[3] / (6*Pi)^(2*k-1) - (9^k-3)*Zeta'[-2*k+1]/2/9^k], 120]

zetahurwitz'(-1, 2/3) \\ Michel Marcus, Mar 24 2019

Equals Pi/(18*sqrt(3)) - log(3)/72 - PolyGamma(1, 1/3) / (12*sqrt(3)*Pi) - Zeta'(-1)/3.

A324993 + A324994 = -log(3)/36 - 2*Zeta'(-1)/3.

cp's OEIS Frontend

A306651 a(n) = Product_{k=1..n} BarnesG(3*k).

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