A272094 a(n) = Product_{k=0..n} binomial(k^2,k).

1, 1, 6, 504, 917280, 48735086400, 94925811409228800, 8154182636726616909619200, 36091760791026276649159689107865600, 9415901310649088228943246038670339934863360000, 162992165498634702043940163611264755298214594247272038400000

Offset: 0

Author

Vaclav Kotesovec, Apr 20 2016

Keywords

• nonn
• easy

Crossrefs

Cf. A255322, A272093, A272095.

Cf. A000178, A098694, A268196, A262261.

Programs

Mathematica

Table[Product[Binomial[k^2, k], {k, 0, n}], {n, 0, 10}]

Formula

a(n) = A255322(n) / (A272168(n) * A000178(n)).

a(n) ~ c1/c2 * A * exp(-1/12 + n/2 + n^2/4) * n^(1/12 + n^2/2) / (2*Pi)^(n/2), where c1 = Product_{k>=1} (k^2)!/stirling(k^2) = 1.14426047263759216966268786..., c2 = Product_{k>=2} (k*(k-1))!/stirling(k*(k-1)) = 1.086533635964823338078329..., stirling(n) = sqrt(2*Pi*n) * n^n / exp(n) is the Stirling approximation of n!, and A = A074962 is the Glaisher-Kinkelin constant.