A372140 - OEIS

%I #7 Apr 20 2024 23:58:51

%S 1,1,1,1,16,3981312,2271857773302207479808,

%T 133781874275586180035265927852035878702421114880000000

%N a(n) = Product_{k=1..n} BarnesG(k)^k.

%C The next term has 113 digits.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/BarnesG-Function.html">Barnes G-Function</a>.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Barnes_G-function">Barnes G-function</a>.

%F a(n) ~ (2*Pi)^(n*(n^2 - 1)/6) * n^(n^4/8 - n^3/12 - n^2/6 + n/24 + 19/720) / (A^(n^2/2 + n/2 - 1/3) * exp(7*n^4/32 - 5*n^3/72 - 7*n^2/24 - n/24 - zeta(3)/(8*Pi^2) + zeta'(-3)/6 + 23/720)), where A is the Glaisher-Kinkelin constant A074962, zeta(3) = A002117, zeta'(-3) = A259068.

%t Table[Product[BarnesG[k]^k, {k, 1, n}], {n, 0, 8}]

%Y Cf. A055462, A255269.

%K nonn

%O 0,5

%A _Vaclav Kotesovec_, Apr 20 2024