OFFSET
0,2
COMMENTS
In general, for m > 1, Product_{k=1..n} binomial(n + 1/k^m, n) ~ n^Zeta(m) / c(m), where c(m) = Product_{j>=1} Gamma(1 + 1/j^m)).
Equivalently, c(m) = -gamma * Zeta(m) + Sum_{k>=2} (-1)^k*Zeta(k)*Zeta(m*k)/k, where gamma is the Euler-Mascheroni constant A001620.
FORMULA
a(n) ~ n!^(4*n) * n^Zeta(3) / (Product_{j>=1} Gamma(1 + 1/j^3)).
a(n) ~ n^(4*n^2 + 2*n + Zeta(3)) * (2*Pi)^(2*n) / exp(4*n^2 - 1/3 - gamma*Zeta(3) + c), where c = A306778 = Sum_{k>=2} (-1)^k*Zeta(k)*Zeta(3*k)/k.
MAPLE
a:= n-> n!^(4*n)*mul(binomial(n+1/k^3, n), k=1..n):
seq(a(n), n=0..7); # Alois P. Heinz, Jun 24 2023
MATHEMATICA
Table[n!^(4*n) * Product[Binomial[n + 1/j^3, n], {j, 1, n}], {n, 1, 8}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Vaclav Kotesovec, Mar 09 2019
EXTENSIONS
a(0)=1 prepended by Alois P. Heinz, Jun 24 2023
STATUS
approved