A296589 a(n) = Product_{k=0..n} binomial(2*n, k).

1, 2, 24, 1800, 878080, 2857680000, 63117561830400, 9577928124440387712, 10077943267571584204800000, 74054886893191804566576837427200, 3822038592032831128918160803430400000000, 1391938996758770867922655936144556115037409280000

Offset: 0

Author

Vaclav Kotesovec, Dec 16 2017

Keywords

• nonn

Links

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

Crossrefs

Cf. A001142, A007685, A086205, A110131, A112332, A268196, A296590, A296591.

Programs

Mathematica

Table[Product[Binomial[2*n, k], {k, 0, n}], {n, 0, 12}]
Table[((2*n)!)^(n+1) / (n! * BarnesG[2*n + 2]), {n, 0, 12}]

Formula

a(n) = ((2*n)!)^(n+1) / (n! * BarnesG(2*n + 2)).

a(n) ~ A * exp(n^2 + n - 1/24) / (2^(5/12) * Pi^((n+1)/2) * n^(n/2 + 5/12)), where A is the Glaisher-Kinkelin constant A074962.

Extensions

Missing a(0)=1 inserted by Georg Fischer, Nov 18 2021