A296591 a(n) = Product_{k=0..n} (n + k)!.

1, 2, 288, 12441600, 421382062080000, 23120161750363668480000000, 3683853104727992382799761899520000000000, 2777528195026874073410445622205453260145295360000000000000

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, A098694, A110131, A112332, A268196, A296589, A296590.

Programs

Maple

a:= proc(n) option remember; `if`(n=0, 1,
      a(n-1) *(2*n-1)! *(2*n)! /(n-1)!)
    end:
seq(a(n), n=0..7);  # Alois P. Heinz, Jul 11 2024

Mathematica

Table[Product[(n + k)!, {k, 0, n}], {n, 0, 10}]
Table[Product[(2*n - k)!, {k, 0, n}], {n, 0, 10}]
Table[BarnesG[2*n + 2]/BarnesG[n + 1], {n, 0, 10}]

Formula

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

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).

Extensions

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