A306789 a(n) = Product_{k=0..n} binomial(n + k, n).

1, 2, 18, 800, 183750, 224042112, 1475939646720, 53195808994099200, 10587785727897772143750, 11721562427290210695200000000, 72596493516095364770534596279431168, 2527156530619699341247423878706695556300800, 496395279097923766533851314609410101501472675840000

Offset: 0

Author

Vaclav Kotesovec, Mar 10 2019

Keywords

• nonn

Comments

Sum_{k=0..n} binomial(n + k, n) = binomial(2*n + 1, n).

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

Crossrefs

Cf. A001700, A306760.

Programs

Mathematica

Table[Product[Binomial[n+k, n], {k, 0, n}], {n, 0, 13}]
Table[(n+1)^n * BarnesG[2*n+2] / (Gamma[n+2]^n * BarnesG[n+2]^2), {n, 0, 13}]

Python

from math import factorial
from functools import lru_cache
@lru_cache(maxsize=None)
def A306789(n): return A306789(n-1)*2*n*factorial(2*n-1)**2//factorial(n)**3//n**(n-1) if n else 1 # Chai Wah Wu, Jun 26 2023

Formula

a(n) = (n+1)^n * BarnesG(2*n+2) / (Gamma(n+2)^n * BarnesG(n+2)^2).

a(n) ~ A * 2^(2*n^2 + 3*n/2 - 1/12) / (exp(n^2/2 + 1/6) * Pi^((n+1)/2) * n^(n/2 + 5/12)), where A is the Glaisher-Kinkelin constant A074962.

a(n) = a(n-1)*2n*(2n-1)!^2/(n!^3*n^(n-1)). - Chai Wah Wu, Jun 26 2023