cp's OEIS Frontend

Equals limit n->infinity (Product_{k=0..n} (k^2)!) / (n^((2*n + 1)*(2*n^2 + 2*n + 3)/6) * (2*Pi)^(n/2) / exp(5*n^3/9 + n^2/2 + n)).

Equals sqrt(2*Pi) * exp(Zeta(3)/(2*Pi^2)) * Product_{n>=1} ((n^2)!/stirling(n^2)), where stirling(n^2) = sqrt(2*Pi) * n^(2*n^2+1) / exp(n^2) is the Stirling approximation of (n^2)!. - Vaclav Kotesovec, Apr 20 2016

a(n) ~ c * n^(1 + 28*n/15 + 4*n^3/3 + 2*n^4 + 4*n^5/5) * (2*Pi)^(n/2) / exp(19*n/9 + n^4/2 + 9*n^5/25), where c = A255438 = 6.644987918706354049483118... .

a(n) = Product_{j=1..n^4} j^(n - ceiling(j^(1/4)) + 1). - Vaclav Kotesovec, Apr 25 2024

Equals limit n->infinity (Product_{k=0..n} (k^5)!) / (n^(80/63 + 5*n/2 - 5*n^2/12 + 25*n^4/12 + 5*n^5/2 + (5*n^6)/6) * (2*Pi)^(n/2) / exp(5*n/2 + 35*n^2/144 + n^5/2 + 11*n^6/36)).

Equals 2^(5/4)*Pi^(5/4)*exp(137/3024 - 5*Zeta'(-5)) * Product_{n>=1} ((n^5)! / stirling(n^5)), where stirling(n^5) = sqrt(2*Pi) * n^(5*n^5 + 5/2) / exp(n^5) is the Stirling approximation of (n^5)! and Zeta'(-5) = A259070. - Vaclav Kotesovec, Apr 20 2016

Equals limit n->infinity (Product_{k=0..n} (k^3)!) / (n^(29/40 + 3*n/2 + 3*n^2/4 + 3*n^3/2 + 3*n^4/4) * (2*Pi)^(n/2) / exp(n*(n+2)*(12 - 6*n + 7*n^2)/16)).

Equals (2*Pi)^(3/4) * exp(-11/240 - 3*Zeta'(-3)) * Product_{n>=1} ((n^3)!/stirling(n^3)), where stirling(n^3) = sqrt(2*Pi) * n^(3*n^3 + 3/2) / exp(n^3) is the Stirling approximation of (n^3)! and Zeta'(-3) = A259068. - Vaclav Kotesovec, Apr 20 2016