OFFSET
0,2
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.
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
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}]
PROG
(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
CROSSREFS
KEYWORD
nonn
AUTHOR
Vaclav Kotesovec, Mar 10 2019
STATUS
approved