OFFSET
0,3
COMMENTS
LINKS
Eric Weisstein's World of Mathematics, Barnes G-Function.
Wikipedia, Barnes G-function
FORMULA
a(n) = A^(3/2) * 2^(n^2 - 1/24) * BarnesG(n + 3/2) / (exp(1/8) * Pi^(n/2 + 1/4) * BarnesG(n + 2)), where A is the Glaisher-Kinkelin constant A074962.
a(n) ~ A^(3/2) * exp(n/2 - 1/8) * 2^(n^2 - 7/24) / (Pi^((n+1)/2) * n^(n/2 + 3/8)), where A is the Glaisher-Kinkelin constant A074962.
Product_{1 <= j <= i <= n} (i + j - 1)/(i - j + 1). - Peter Bala, Oct 25 2024
MAPLE
A296590 := proc(n)
mul( binomial(2*n-k, k), k=0..n) ;
end proc:
seq(A296590(n), n=0..7) ; # R. J. Mathar, Jan 03 2018
MATHEMATICA
Table[Product[Binomial[2*n-k, k], {k, 0, n}], {n, 0, 15}]
Table[Glaisher^(3/2) * 2^(n^2 - 1/24) * BarnesG[n + 3/2] / (E^(1/8) * Pi^(n/2 + 1/4) * BarnesG[n + 2]), {n, 0, 15}]
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Vaclav Kotesovec, Dec 16 2017
STATUS
approved