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 A385438 #75 Aug 04 2025 04:26:25
%S A385438 1,10,208,4888,121132,3092950,80506684,2123780536,56581885468,
%T A385438 1518936682888,41021505946468,1113273696074968,30335161535834212,
%U A385438 829405495046080612,22742967214283811976,625193974445825554408,17223870801864911429404,475423918887141016417144
%N A385438 a(n) = Sum_{k=0..n} 3^k * binomial(3*n+k-1,k).
%H A385438 Seiichi Manyama, <a href="/A385438/b385438.txt">Table of n, a(n) for n = 0..689</a>
%F A385438 a(n) = Sum_{k=0..n} 2^(n-k) * binomial(4*n,k) * binomial(4*n-k-1,n-k).
%F A385438 a(n) = [x^n] ( (1+x)^4/(1-2*x)^3 )^n.
%F A385438 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)^3 / (1+x)^4 ). See A385475.
%F A385438 a(n) = Sum_{k=0..n} 3^k * (-2)^(n-k) * binomial(4*n,k).
%F A385438 a(n) = (-2)^(-3*n) - 3^n*binomial(4*n-1, n)*(hypergeom([1, 4*n], [1+n], 3) - 1). - _Stefano Spezia_, Aug 02 2025
%F A385438 a(n) ~ 2^(8*n + 1/2) / (11 * 3^(2*n - 3/2) * sqrt(Pi*n)). - _Vaclav Kotesovec_, Aug 04 2025
%o A385438 (PARI) a(n) = sum(k=0, n, 3^k*binomial(3*n+k-1, k));
%Y A385438 Cf. A383888, A384950.
%Y A385438 Cf. A385475.
%K A385438 nonn
%O A385438 0,2
%A A385438 _Seiichi Manyama_, Aug 01 2025