A255674 -id:A255674 - cp's OEIS frontend

A306635 a(n) = Product_{k=1..n} BarnesG(2*k).

Original entry on oeis.org

1, 2, 576, 14332723200, 72474629486854275072000000, 482580045081719158086051946616717605601280000000000000

Offset: 1

Vaclav Kotesovec, Mar 02 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, A168467, A255674, A296607, A306651, A324992.

Table[Product[BarnesG[2*k], {k, 1, n}], {n, 1, 8}]
Round[Table[2^(2*n^3/3 + n^2 - 5*n/3 - 2/3) * E^(n^3/2 + 3*n^2/4 + n/4 + 1/12 - 3*Zeta[3]/(16*Pi^2) + 2*PolyGamma[-3, n + 1] + Derivative[1, 0][Zeta][-2, n + 1/2] + 2*Derivative[1, 0][Zeta][-1, n + 1/2]) * Gamma[n]^(2*n - 7/4) * BarnesG[2*n]^(n + 1) / (Glaisher^(2*n + 3) * Pi^(n^2/2 + n + 1/2) * n^(n^2) * Gamma[2*n]^(n^2 + n - 3/4) * BarnesG[n]^2), {n, 1, 8}]] (* Vaclav Kotesovec, Mar 04 2019 *)

a(n) ~ c * 2^(2*n^3/3 + n^2/2 - n/4 - 3/8) * n^(2*n^3/3 - n/4) * Pi^(n^2/2 - 3/8) / (A^(n-2) * exp(11*n^3/9 - n/3 - Zeta(3)/(2*Pi^2) + 1/12)), where c = A255674^2 = 1.1446513373245340524595435844492841792576337833610236993... and A is the Glaisher-Kinkelin constant A074962.
a(n) ~ 2^(2*n^3/3 + n^2/2 - n/4 - 1/8) * n^(2*n^3/3 - n/4) * Pi^(n^2/2) / (A^n * exp(11*n^3/9 - n/3 - Zeta(3)/(16*Pi^2))), where A is the Glaisher-Kinkelin constant A074962.
a(n) = a(n-1)*A296607(n). - R. J. Mathar, Jul 24 2025

A055746 Product of first n terms of A003046.

Original entry on oeis.org

1, 1, 2, 20, 2800, 16464000, 12778698240000, 4254956888736153600000, 2026001446509988558521630720000000, 4690285643617101997210180025102660272128000000000

Offset: 0

N. J. A. Sloane, Jul 11 2000

nonn

Cf. A000108, A003046, A000984, A055462, A255674, A306635.

seq(mul(mul(binomial(2*j,j)/(j+1),j=0..k), k=0..n), n=0..9); # Zerinvary Lajos, Sep 21 2007

Table[Product[Product[Binomial[2*j,j]/(j+1),{j,0,k}],{k,0,n}],{n,0,10}] (* Vaclav Kotesovec, Jul 10 2015 *)
Table[Product[2^((k + 1)/2) * Sqrt[BarnesG[2*k]] * Gamma[2*k] / (BarnesG[k] * BarnesG[k + 3] * Gamma[k]^(3/2)), {k, 1, n}], {n, 0, 10}] (* Vaclav Kotesovec, Mar 02 2019 *)

a(n) ~ c * 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)), where A = A074962 = 1.2824271291006226368753425688697917277... is the Glaisher-Kinkelin constant and c = 1.06988379617813356826829257647028132359737354153723273083785714620398... = A255674. - Vaclav Kotesovec, Jul 10 2015
a(n) ~ A^(3*n/2 + 3) * exp(9*n^2/8 + 5*n/2 - 7*Zeta(3)/(32*Pi^2) - 1/4) * 2^(n^3/3 + n^2 - n/8 - 65/48) / (Pi^(n^2/4 + 5*n/4 + 3/2) * n^(3*n^2/4 + 21*n/8 + 9/4)), where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, Mar 02 2019
a(n) = Product_{k=1..n} (2^((k+1)/2) * sqrt(BarnesG(2*k)) * Gamma(2*k) / (BarnesG(k) * BarnesG(k+3) * Gamma(k)^(3/2))). - Vaclav Kotesovec, Mar 02 2019

cp's OEIS Frontend

A306635 a(n) = Product_{k=1..n} BarnesG(2*k).

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