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 A384950 #31 Aug 04 2025 04:21:24
%S A384950 1,7,103,1720,30319,550867,10204660,191606380,3633593071,69434167357,
%T A384950 1334845289023,25787841299392,500217562201348,9736067678711524,
%U A384950 190051513661403112,3719197868485767940,72942019051301120239,1433317465944902210161,28212929859612197439829
%N A384950 a(n) = Sum_{k=0..n} 3^k * binomial(2*n+k-1,k).
%H A384950 Seiichi Manyama, <a href="/A384950/b384950.txt">Table of n, a(n) for n = 0..766</a>
%F A384950 a(n) = Sum_{k=0..n} 2^(n-k) * binomial(3*n,k) * binomial(3*n-k-1,n-k).
%F A384950 a(n) = [x^n] ( (1+x)^3/(1-2*x)^2 )^n.
%F A384950 The g.f. exp( Sum_{k>=1} a(k) * x^k/k ) has integer coefficients and equals (1/x) * Series_Reversion( x * (1-2*x)^2 / (1+x)^3 ). See A385474.
%F A384950 a(n) = Sum_{k=0..n} 3^k * (-2)^(n-k) * binomial(3*n,k).
%F A384950 a(n) = 4^(-n) - 3^n*binomial(3*n-1, n)*(hypergeom([1, 3*n], [1+n], 3) - 1). - _Stefano Spezia_, Aug 02 2025
%F A384950 a(n) ~ 3^(4*n + 3/2) / (2^(2*n+3) * sqrt(Pi*n)). - _Vaclav Kotesovec_, Aug 04 2025
%o A384950 (PARI) a(n) = sum(k=0, n, 3^k*binomial(2*n+k-1, k));
%Y A384950 Cf. A383888, A385438.
%Y A384950 Cf. A385474.
%K A384950 nonn
%O A384950 0,2
%A A384950 _Seiichi Manyama_, Aug 01 2025