A371646 -id:A371646 - cp's OEIS frontend

A371603 a(n) = Product_{k=0..n} binomial(n^2, k^2).

1, 1, 4, 1134, 333132800, 1319947441510156250, 876533819183888230348458418944000, 1185269534290897564185384010731432113450477770983533184

Offset: 0

Vaclav Kotesovec, Mar 30 2024

nonn

Cf. A001142, A255322, A272095, A296591, A362288, A371624, A371646.

Table[Product[Binomial[n^2, k^2], {k, 0, n}], {n, 0, 8}]

a(n) = (n^2)!^(n+1) / (A255322(n) * A371624(n)).

a(n) ~ c * exp(2*n*(2*n^2/3 + 1)) / (A^(2*n) * 2^(4*n*(n^2 + 1)/3) * Pi^(n/2) * n^(7*n/6 - 1/4)), where c = 0.6367427... and A is the Glaisher-Kinkelin constant A074962.

A371468 a(n) = Product_{k=0..n} (n^3 + k^3)!.

Original entry on oeis.org

1, 2, 306128067620555980800000

Offset: 0

Vaclav Kotesovec, Apr 01 2024

nonn

The next term a(3) has 169 digits.

Cf. A296591, A371643, A371646.

Table[Product[(n^3 + k^3)!, {k, 0, n}], {n, 0, 5}]

a(n) ~ 2^(2*n^4 + n^3 + n^2/4 + 3*n/2 + 89/120) * Pi^((n+1)/2) * exp(Pi*sqrt(3)*n^4/4 - 59*n^4/16 - 3*n^3/2 - 3*n/2 + Pi*n/(2*sqrt(3)) - 9/320) * n^(15*n^4/4 + 9*n^3/2 + 3*n^2/4 + 3*n/2 + 3/2).

cp's OEIS Frontend

A371603 a(n) = Product_{k=0..n} binomial(n^2, k^2).

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