A372117 a(n) = Product_{k=0..n} binomial(n+k, k)^k.

1, 2, 108, 3200000, 1158107343750000, 119025168578031262646195453952, 82864944710388642300699757862681018776776867840000, 9481019710293786574190900386319772308050021208649248212215823364196925440000000

Offset: 0

Author

Vaclav Kotesovec, Apr 19 2024

Keywords

• nonn

Crossrefs

Cf. A067055, A255269, A306789, A362288, A372116.

Programs

Mathematica

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

Formula

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

a(n) = A372116(n) / (A255269(n) * A067055(n)).

a(n) ~ 2^(2*n^3/3 + 3*n^2/4 + n/6 + 1/24) * exp(n^3/12 + n^2/4 - n/24 + zeta(3)/(8*Pi^2) - 1/24) / (sqrt(A) * Pi^(n^2/4 + n/4) * n^(n^2/4 + n/4 + 1/24)), where A is the Glaisher-Kinkelin constant A074962.