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

0,2

Links

Table of n, a(n) for n=0..22.

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

Cf. A073165, A120259.

Sequence in context: A131351 A091352 A135934 * A210342 A137154 A098448

Adjacent sequences: A343029 A343030 A343031 * A343033 A343034 A343035

Keyword

nonn

Author

Seiichi Manyama, Apr 03 2021

Status

approved