OFFSET
0,2
FORMULA
a(n) = Sum_{k=0..n} Product_{1<=i<=j<=k} (n-k+i+j-1)/(i+j-1).
Limit_{n->infinity} a(n)^(1/n^2) = 2^r * r^(r/2) * (1-r)^((1-r)/2) = 1.113022855718664043805172905388731078607920794227951582456470883692074109..., where r = 0.62986938372832785012478891433662812255632994055776040984266... is the root of the equation 2^(4*r) * (1-r)^(1-r) * r^(2*r) = (1+r)^(1+r). - Vaclav Kotesovec, Apr 03 2021
MATHEMATICA
Table[Sum[Product[(n - k + i + j - 1)/(i + j - 1), {i, 1, k}, {j, 1, i}], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Apr 03 2021 *)
Table[Sum[BarnesG[k+1] / BarnesG[n+1] * Sqrt[Gamma[k+1] * Gamma[(n-k+2)/2] * BarnesG[n-k+1] * BarnesG[n+k+2] / (Gamma[n-k+1] * Gamma[(n+k+2)/2] * BarnesG[2*k+2])], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Apr 03 2021 *)
PROG
(PARI) a(n) = sum(k=0, n, prod(i=1, k, prod(j=1, i, (n-k+i+j-1)/(i+j-1))));
CROSSREFS
KEYWORD
nonn
AUTHOR
Seiichi Manyama, Apr 03 2021
STATUS
approved