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 A349533 #15 Aug 01 2023 09:33:42
%S A349533 1,3,13,74,499,3719,29494,243888,2078431,18122369,160885449,
%T A349533 1449268478,13213370392,121696581804,1130565483472,10581614352704,
%U A349533 99685591788687,944490400760597,8994266558594671,86040075341770806,826423263373253923,7967095415955791687
%N A349533 G.f. A(x) satisfies A(x) = 1 / ((1 - 2 * x) * (1 - x * A(x)^2)).
%H A349533 Seiichi Manyama, <a href="/A349533/b349533.txt">Table of n, a(n) for n = 0..989</a>
%F A349533 a(n) = 2^n + Sum_{i=0..n-1} Sum_{j=0..n-i-1} a(i) * a(j) * a(n-i-j-1).
%F A349533 a(n) = Sum_{k=0..n} binomial(n+k,2*k) * binomial(3*k,k) * 2^(n-k) / (2*k+1).
%F A349533 a(n) = 2^n*F([1/3, 2/3, -n, 1+n], [1/2, 1, 3/2], -3^3/2^5), where F is the generalized hypergeometric function. - _Stefano Spezia_, Nov 21 2021
%F A349533 a(n) ~ 177^(1/4) * (43 + 3*sqrt(177))^(n + 1/2) / (9 * sqrt(Pi) * n^(3/2) * 2^(3*n + 5/2)). - _Vaclav Kotesovec_, Nov 22 2021
%t A349533 nmax = 21; A[_] = 0; Do[A[x_] = 1/((1 - 2 x) (1 - x A[x]^2)) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
%t A349533 a[n_] := a[n] = 2^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, 21}]
%t A349533 Table[Sum[Binomial[n + k, 2 k] Binomial[3 k, k] 2^(n - k)/(2 k + 1), {k, 0, n}], {n, 0, 21}]
%Y A349533 Cf. A001764, A064613, A199475, A349253, A349534, A349535.
%K A349533 nonn
%O A349533 0,2
%A A349533 _Ilya Gutkovskiy_, Nov 21 2021