A120260 Diagonal sums of number triangle A120258.
1, 1, 2, 3, 8, 24, 92, 432, 2740, 23822, 264185, 3545166, 59474514, 1343942004, 41179884383, 1593533376361, 74665098131246, 4404743069577837, 351138858279113987, 37740395752334771775, 5093113605218543006445
Offset: 0
Programs
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Mathematica
Table[Sum[Product[Binomial[2*n-4*k+j, n-2*k]/Binomial[n-2*k+j, j], {j,0,k-1}], {k,0,Floor[n/2]}], {n,0,20}] (* Vaclav Kotesovec, Aug 29 2023 *) Table[Sum[BarnesG[1 + k] * BarnesG[2 - 2*k + n]^2 * BarnesG[1 - 3*k + 2*n] * Gamma[1 - 4*k + 2*n] / (BarnesG[1 - k + n]^2 * BarnesG[2 - 4*k + 2*n] * Gamma[1 - 2*k + n]^2), {k, 0, Floor[n/2]}], {n, 0, 20}] (* Vaclav Kotesovec, Aug 29 2023 *)
Formula
a(n)=sum{k=0..floor(n/2), Product{j=0..k-1, C(2n-4k+j, n-2k)/C(n-2k+j, j)}}
Limit_{n->oo} a(n)^(1/n^2) = r^(r^2/2) * (2-3*r)^((2-3*r)^2/2) / (2^(2*(1-2*r)^2) * (1-r)^((1-r)^2) * (1-2*r)^((1-2*r)^2)) = 1.133380884076924860904704854418..., where r = 0.201760656726887011996310570327419178... is the root of the equation 2^(8-16*r) * (2-3*r)^(-6+9*r) * (1-2*r)^(4-8*r) * (1-r)^(2-2*r) * r^r = 1. - Vaclav Kotesovec, Aug 29 2023