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 A344500 #12 Apr 21 2024 04:40:45
%S A344500 1,1,2,7,21,66,216,715,2395,8101,27598,94568,325612,1125632,3904512,
%T A344500 13583195,47373255,165585883,579907758,2034443127,7148313381,
%U A344500 25151582046,88607951512,312518438532,1103393962996,3899415207676,13792683831176,48825746365672,172971084083752
%N A344500 a(n) = Sum_{k=0..n} binomial(n, k)*CT(n, k) = Sum_{k=0..n} (-1)^(n-k)*binomial(n, k)*CT(n, k), where CT(n, k) is the Catalan triangle A053121.
%F A344500 a(n) = Sum_{j=0..n} even(n + j)*binomial(n, j)*binomial(n + 1, (n - j)/2)*(j + 1)/(n + 1), where even(k) = 1 if k is even and otherwise 0.
%F A344500 From _Vaclav Kotesovec_, Apr 21 2024: (Start)
%F A344500 Recurrence: 2*(n+1)*(2*n + 1)*(13*n^3 - 17*n^2 - 12*n + 10)*a(n) = (143*n^5 - 44*n^4 - 525*n^3 + 320*n^2 + 94*n - 60)*a(n-1) + 4*(n-1)*(26*n^4 + 5*n^3 - 108*n^2 + 7*n + 22)*a(n-2) + 16*(n-2)*(n-1)*(13*n^3 + 22*n^2 - 7*n - 6)*a(n-3).
%F A344500 a(n) ~ sqrt((247 - 9131*13^(2/3)/(1788163 + 409728*sqrt(78))^(1/3) + (13*(1788163 + 409728*sqrt(78)))^(1/3))/13) * ((11 + (1/3)*(172017 - 16848*sqrt(78))^(1/3) + (6371 + 624*sqrt(78))^(1/3))^n / (sqrt(Pi*n) * 2^(2*n + 3) * 3^(n + 1/2))). (End)
%p A344500 a := n -> add((if n + j mod 2 = 1 then 0 else binomial(n, j)*binomial(n + 1, (n - j)/2)*(j + 1)/(n + 1) fi), j = 0..n): seq(a(n), n = 0..28);
%t A344500 Table[Sum[(1 + (-1)^(n+j))/2 * Binomial[n, j] * Binomial[n+1, (n-j)/2] * (j+1)/(n+1), {j, 0, n}], {n, 0, 30}] (* _Vaclav Kotesovec_, Apr 21 2024 *)
%Y A344500 Cf. A053121, A344501.
%K A344500 nonn
%O A344500 0,3
%A A344500 _Peter Luschny_, May 22 2021