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 A349534 #15 Aug 01 2023 09:33:39
%S A349534 1,4,21,138,1063,9075,82770,789204,7766721,78267306,803447526,
%T A349534 8371413999,88300495746,941004684748,10116276976218,109578418285452,
%U A349534 1194764348642313,13102287157827918,144422108994233625,1599198859915070640,17780781456147340764
%N A349534 G.f. A(x) satisfies A(x) = 1 / ((1 - 3 * x) * (1 - x * A(x)^2)).
%H A349534 Seiichi Manyama, <a href="/A349534/b349534.txt">Table of n, a(n) for n = 0..931</a>
%F A349534 a(n) = 3^n + Sum_{i=0..n-1} Sum_{j=0..n-i-1} a(i) * a(j) * a(n-i-j-1).
%F A349534 a(n) = Sum_{k=0..n} binomial(n+k,2*k) * binomial(3*k,k) * 3^(n-k) / (2*k+1).
%F A349534 a(n) = 3^n*F([1/3, 2/3, -n, 1+n], [1/2, 1, 3/2], -3^2/2^4), where F is the generalized hypergeometric function. - _Stefano Spezia_, Nov 21 2021
%F A349534 a(n) ~ sqrt(5/Pi) * 3^(n-1) * 4^n / n^(3/2). - _Vaclav Kotesovec_, Nov 22 2021
%t A349534 nmax = 20; A[_] = 0; Do[A[x_] = 1/((1 - 3 x) (1 - x A[x]^2)) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
%t A349534 a[n_] := a[n] = 3^n + Sum[Sum[a[i] a[j] a[n - i - j - 1], {j, 0, n - i - 1}], {i, 0, n - 1}]; Table[a[n], {n, 0, 20}]
%t A349534 Table[Sum[Binomial[n + k, 2 k] Binomial[3 k, k] 3^(n - k)/(2 k + 1), {k, 0, n}], {n, 0, 20}]
%Y A349534 Cf. A001764, A104455, A199475, A349254, A349533, A349535.
%K A349534 nonn
%O A349534 0,2
%A A349534 _Ilya Gutkovskiy_, Nov 21 2021