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 A378565 #9 Dec 01 2024 05:24:33
%S A378565 1,1,7,43,271,1746,11425,75615,504799,3392953,22930282,155664356,
%T A378565 1060710457,7250779238,49700101101,341474150583,2351032782783,
%U A378565 16216401440106,112035931072915,775163096510445,5370301986029066,37249469056575504,258648802856972348
%N A378565 a(n) = Sum_{k=0..n} binomial(n+k-1,k) * binomial(n+k-1,n-k).
%F A378565 a(n) = [x^n] 1/(1 - x/(1 - x)^2)^n.
%F A378565 a(n) ~ (525 - 32*210^(2/3)/(157*sqrt(105) - 1575)^(1/3) + 4*(210*(157*sqrt(105) - 1575))^(1/3))^(1/6) * ((36 + (1208682 - 28350*sqrt(105))^(1/3)/3 + (6*(7461 + 175*sqrt(105)))^(1/3))^n / (2^(2/3) * 7^(1/3) * sqrt(Pi*n) * 3^(n + 1/6) * 5^(n + 1/3))). - _Vaclav Kotesovec_, Dec 01 2024
%t A378565 Table[Sum[Binomial[n+k-1, k] * Binomial[n+k-1, 2*k-1], {k, 0, n}], {n, 0, 25}] (* _Vaclav Kotesovec_, Dec 01 2024 *)
%o A378565 (PARI) a(n) = sum(k=0, n, binomial(n+k-1, k)*binomial(n+k-1, n-k));
%Y A378565 Cf. A002002, A378566, A378567.
%Y A378565 Cf. A011270, A362087.
%K A378565 nonn
%O A378565 0,3
%A A378565 _Seiichi Manyama_, Dec 01 2024