A362288 a(n) = Product_{k=0..n} binomial(n,k)^k.

1, 1, 2, 27, 9216, 312500000, 4251528000000000, 95432797246104853383515625, 14719075154533285649961930052505436160000, 65577306173662530591576256095315195684570038194755952705536

Offset: 0

Author

Vaclav Kotesovec, Apr 14 2023

Keywords

• nonn

Links

• Winston de Greef, Table of n, a(n) for n = 0..22

Crossrefs

Cf. A001142, A067055, A167008, A255268, A272093.

Programs

Mathematica

Table[Product[Binomial[n, k]^k, {k, 0, n}], {n, 0, 10}]
Table[(n!)^(n*(n+1)/2) / BarnesG[n+2]^n, {n, 0, 10}]

PARI

a(n) = prod(k=0, n, binomial(n,k)^k); \\ Michel Marcus, Apr 14 2023

Formula

a(n) = Product_{k=0..n} n!^k / k!^n.

a(n) = A067055(n) / A255268(n).

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