A343032 Row sums of triangle A073165.
1, 2, 4, 9, 24, 78, 313, 1557, 9606, 73482, 696736, 8187149, 119214337, 2150935400, 48085463503, 1331903411529, 45708405952786, 1943464419169294, 102378212255343442, 6681679619583450775, 540264005909352759970, 54120992439329583459008, 6716802027097934788929023
Offset: 0
Keywords
Programs
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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 *) -
PARI
a(n) = sum(k=0, n, prod(i=1, k, prod(j=1, i, (n-k+i+j-1)/(i+j-1))));
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