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 A378405 #9 Nov 25 2024 07:35:39
%S A378405 1,1,3,13,47,171,651,2507,9703,37831,148393,584673,2312267,9174179,
%T A378405 36500257,145566333,581746503,2329206823,9341025429,37516150599,
%U A378405 150874376997,607479424817,2448608334087,9879562243961,39897969991075,161260133795371
%N A378405 a(n) = Sum_{k=0..floor(n/2)} binomial(n,k) * binomial(n+k,n-2*k).
%F A378405 a(n) = [x^n] (1 + x + x^2 * (1 + x)^2)^n.
%F A378405 a(n) ~ sqrt(c) * d^n / sqrt(Pi*n), where d = 4.1236218756427610008124277125077732535524468472302771364162049292... is the greatest root of the equation 31 - 256*d + 30*d^2 - 104*d^3 + 27*d^4 = 0 and c = 0.3580097056143148810957018324419771260252367819271313726816... is the positive real root of the equation -31 - 3024*c + 51376*c^2 - 257536*c^3 + 386304*c^4 = 0. - _Vaclav Kotesovec_, Nov 25 2024
%t A378405 Table[Sum[Binomial[n, k]*Binomial[n + k, n - 2*k], {k, 0, n/2}], {n, 0, 30}] (* _Vaclav Kotesovec_, Nov 25 2024 *)
%o A378405 (PARI) a(n) = sum(k=0, n\2, binomial(n, k)*binomial(n+k, n-2*k));
%Y A378405 Cf. A005725, A378406.
%Y A378405 Cf. A001850, A082759.
%K A378405 nonn
%O A378405 0,3
%A A378405 _Seiichi Manyama_, Nov 25 2024