A055746 Product of first n terms of A003046.

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

Offset

0,3

Links

Table of n, a(n) for n=0..9.

Formula

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

Maple

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

Mathematica

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 *)

Crossrefs

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

Sequence in context: A319639 A134476 A224732 * A258878 A060600 A356689

Adjacent sequences: A055743 A055744 A055745 * A055747 A055748 A055749

Keyword

nonn

Author

N. J. A. Sloane, Jul 11 2000

Status

approved