This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A120260 #7 Aug 29 2023 04:32:05
%S A120260 1,1,2,3,8,24,92,432,2740,23822,264185,3545166,59474514,1343942004,
%T A120260 41179884383,1593533376361,74665098131246,4404743069577837,
%U A120260 351138858279113987,37740395752334771775,5093113605218543006445
%N A120260 Diagonal sums of number triangle A120258.
%F A120260 a(n)=sum{k=0..floor(n/2), Product{j=0..k-1, C(2n-4k+j, n-2k)/C(n-2k+j, j)}}
%F A120260 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
%t A120260 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 *)
%t A120260 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 *)
%K A120260 easy,nonn
%O A120260 0,3
%A A120260 _Paul Barry_, Jun 13 2006