A324590 a(n) = n!^(4*n) * Product_{k=1..n} binomial(n + 1/k^2, n).

1, 2, 1080, 16133644800, 139256878046022696960000, 6288402750181849898716908922601472000000000, 8322157105451357856813375261666887975745751468393837363200000000000000

Offset: 0

Author

Vaclav Kotesovec, Mar 09 2019

Keywords

• nonn

Crossrefs

Cf. A303670, A306760, A306774, A324589.

Programs

Maple

a:= n-> n!^(4*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!^(4*n) * Product[Binomial[1/k^2 + n, n], {k, 1, n}], {n, 1, 8}]

Formula

a(n) ~ n!^(4*n) * n^(Pi^2/6) / A303670.

a(n) ~ n^(4*n^2 + 2*n + Pi^2/6) * (2*Pi)^(2*n) / exp(4*n^2 - 1/3 - 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.

a(n) = n!^n * A324596(n).

Extensions

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