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 A386865 #10 Aug 06 2025 08:46:51
%S A386865 1,6,51,496,5130,54894,600103,6657312,74646702,843819580,9599776494,
%T A386865 109776491664,1260666279964,14528980409454,167951183468655,
%U A386865 1946529575164864,22611104963042646,263175370423429428,3068541416792813338,35834296592951011680,419059482092284948908
%N A386865 a(n) = Sum_{k=0..n} 2^(n-k) * binomial(2*n+2,k) * binomial(2*n-k-1,n-k).
%F A386865 a(n) = [x^n] (1+x)^(2*n+2)/(1-2*x)^n.
%F A386865 a(n) = [x^n] 1/((1-x)^3 * (1-3*x)^n).
%F A386865 a(n) = Sum_{k=0..n} 3^k * (-2)^(n-k) * binomial(2*n+2,k) * binomial(n-k+2,n-k).
%F A386865 a(n) = Sum_{k=0..n} 3^k * binomial(n+k-1,k) * binomial(n-k+2,n-k).
%F A386865 a(n) ~ 2^(2*n+2) * 3^(n+3) / (125*sqrt(Pi*n)). - _Vaclav Kotesovec_, Aug 06 2025
%t A386865 Table[Sum[(-1)^k*(k+1)*(k+2)*2^(k-1)*3^(n-k)* Binomial[2*n+2, n+k+2], {k, 0, n}], {n, 0, 20}] (* _Vaclav Kotesovec_, Aug 06 2025 *)
%o A386865 (PARI) a(n) = sum(k=0, n, 2^(n-k)*binomial(2*n+2, k)*binomial(2*n-k-1, n-k));
%Y A386865 Cf. A383888, A386862.
%Y A386865 Cf. A386835.
%K A386865 nonn
%O A386865 0,2
%A A386865 _Seiichi Manyama_, Aug 06 2025