A324596 a(n) = n!^(3*n) * Product_{k=1..n} binomial(n + 1/k^2, n).

1, 2, 270, 74692800, 419731620267960000, 252716802910471719823692648960000, 59736659298524125157504488525739821430187940800000000, 16079377413231597423103950774423398920424350187193326745026311068057600000000000

Offset: 0

Author

Vaclav Kotesovec, Mar 09 2019

Keywords

• nonn

Crossrefs

Cf. A303670, A306760, A306774, A324589, A324597.

Programs

Maple

a:= n-> n!^(3*n)*mul(binomial(n+1/k^2, n), k=1..n):
seq(a(n), n=0..7);  # Alois P. Heinz, Jun 24 2023

Mathematica

Table[n!^(3*n) * Product[Binomial[n + 1/k^2, n], {k, 1, n}], {n, 1, 8}]

Formula

a(n) ~ n!^(3*n) * n^(Pi^2/6) / A303670.

a(n) ~ n^(3*n*(2*n+1)/2 + Pi^2/6) * (2*Pi)^(3*n/2) / exp(3*n^2 - 1/4 - gamma*Pi^2/6 + c), where gamma is the Euler-Mascheroni constant A001620 and c = A306774 = Sum_{k>=2} (-1)^k * Zeta(k) * Zeta(2*k) / k.

Extensions

a(0)=1 prepended by Alois P. Heinz, Jun 24 2023