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 A386719 #19 Aug 01 2025 07:49:50
%S A386719 1,11,229,5381,133333,3404156,88600483,2337160718,62263902037,
%T A386719 1671407550260,45137852641204,1224954657942125,33377579214681619,
%U A386719 912572183952374996,25023054179816358034,687862647149533181036,18950129471489195622229,523067259899842250453060
%N A386719 a(n) = Sum_{k=0..n} 2^k * binomial(4*n,k) * binomial(4*n-k-1,n-k).
%F A386719 a(n) = [x^n] ( (1+2*x)^4/(1-x)^3 )^n.
%F A386719 The g.f. exp( Sum_{k>=1} a(k) * x^k/k ) has integer coefficients and equals (1/x) * Series_Reversion( x * (1-x)^3 / (1+2*x)^4 ). See A386723.
%F A386719 a(n) = Sum_{k=0..n} 3^k * (-1)^(n-k) * binomial(4*n,k).
%F A386719 a(n) ~ 2^(8*n - 1/2) / (5 * sqrt(Pi*n) * 3^(2*n - 3/2)). - _Vaclav Kotesovec_, Jul 31 2025
%F A386719 a(n) = Sum_{k=0..n} 3^k * 2^(n-k) * binomial(3*n+k-1,k). - _Seiichi Manyama_, Aug 01 2025
%t A386719 Table[Sum[3^k*(-1)^(n-k)*Binomial[4*n, k], {k, 0, n}], {n, 0, 25}] (* _Vaclav Kotesovec_, Jul 31 2025 *)
%o A386719 (PARI) a(n) = sum(k=0, n, 2^k*binomial(4*n, k)*binomial(4*n-k-1, n-k));
%Y A386719 Cf. A385319, A385320.
%Y A386719 Cf. A386723.
%K A386719 nonn
%O A386719 0,2
%A A386719 _Seiichi Manyama_, Jul 31 2025