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a(n) = Product_{k=0..n} (n + k)!.
9

%I #13 Feb 16 2025 08:33:52

%S 1,2,288,12441600,421382062080000,23120161750363668480000000,

%T 3683853104727992382799761899520000000000,

%U 2777528195026874073410445622205453260145295360000000000000

%N a(n) = Product_{k=0..n} (n + k)!.

%H Eric Weisstein's World of Mathematics, <a href="https://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) = BarnesG(2*n + 2) / BarnesG(n + 1).

%F a(n) ~ 2^(2*n^2 + 5*n/2 + 11/12) * n^((n+1)*(3*n+1)/2) * Pi^((n+1)/2) / exp(9*n^2/4 + 2*n).

%p a:= proc(n) option remember; `if`(n=0, 1,

%p a(n-1) *(2*n-1)! *(2*n)! /(n-1)!)

%p end:

%p seq(a(n), n=0..7); # _Alois P. Heinz_, Jul 11 2024

%t Table[Product[(n + k)!, {k, 0, n}], {n, 0, 10}]

%t Table[Product[(2*n - k)!, {k, 0, n}], {n, 0, 10}]

%t Table[BarnesG[2*n + 2]/BarnesG[n + 1], {n, 0, 10}]

%Y Cf. A001142, A007685, A086205, A098694, A110131, A112332, A268196, A296589, A296590.

%K nonn,changed

%O 0,2

%A _Vaclav Kotesovec_, Dec 16 2017

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