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 A383888 #38 Aug 04 2025 04:17:23
%S A383888 1,4,34,334,3478,37384,409960,4558306,51199558,579554056,6600532684,
%T A383888 75546800476,868224027916,10012494936136,115804853315332,
%U A383888 1342795688895754,15604522381828678,181690692393744376,2119144763079629452,24754486729805925124,289563977079418497748
%N A383888 a(n) = Sum_{k=0..n} 3^k * binomial(n+k-1,k).
%H A383888 Seiichi Manyama, <a href="/A383888/b383888.txt">Table of n, a(n) for n = 0..928</a>
%F A383888 a(n) = Sum_{k=0..n} 2^(n-k) * binomial(2*n,k) * binomial(2*n-k-1,n-k).
%F A383888 a(n) = [x^n] ( (1+x)^2/(1-2*x) )^n.
%F A383888 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) / (1+x)^2 ).
%F A383888 a(n) = Sum_{k=0..n} 3^k * (-2)^(n-k) * binomial(2*n,k).
%F A383888 a(n) = (-2)^(-n)*(1 - (-6)^n*binomial(2*n-1, n)*(hypergeom([1, 2*n], [1+n], 3) - 1)). - _Stefano Spezia_, Aug 02 2025
%F A383888 a(n) ~ 2^(2*n) * 3^(n+1) / (5*sqrt(Pi*n)). - _Vaclav Kotesovec_, Aug 04 2025
%o A383888 (PARI) a(n) = sum(k=0, n, 3^k*binomial(n+k-1,k));
%Y A383888 Cf. A384950, A385438.
%Y A383888 Cf. A064063, A385319.
%K A383888 nonn
%O A383888 0,2
%A A383888 _Seiichi Manyama_, Aug 01 2025