%I #9 Nov 18 2021 19:01:16

%S 1,2,24,1800,878080,2857680000,63117561830400,9577928124440387712,

%T 10077943267571584204800000,74054886893191804566576837427200,

%U 3822038592032831128918160803430400000000,1391938996758770867922655936144556115037409280000

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

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/BarnesG-Function.html">Barnes G-Function</a>.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Barnes_G-function">Barnes G-function</a>

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

%F 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.

%t Table[Product[Binomial[2*n, k], {k, 0, n}], {n, 0, 12}]

%t Table[((2*n)!)^(n+1) / (n! * BarnesG[2*n + 2]), {n, 0, 12}]

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

%K nonn

%O 0,2

%A _Vaclav Kotesovec_, Dec 16 2017

%E Missing a(0)=1 inserted by _Georg Fischer_, Nov 18 2021