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 A349532 #10 Nov 25 2021 03:19:01
%S A349532 1,1,7,52,407,3329,28232,246552,2204895,20103027,186223399,1748009560,
%T A349532 16591329652,158975004204,1535725632552,14940742412112,
%U A349532 146259921123407,1439658075118967,14240062489572485,141469058343614452,1410975387252602527,14122900638031585153
%N A349532 G.f. A(x) satisfies: A(x) = 1 + x * A(x)^3 / (1 - 4 * x).
%C A349532 In general, if k >= 0 and g.f. A(x) satisfies: A(x) = 1 + x * A(x)^3 / (1 - k*x), then a(n) ~ (4*k + 27)^(n + 1/2) / (3 * sqrt(Pi) * n^(3/2) * 4^(n+1)). - _Vaclav Kotesovec_, Nov 25 2021
%F A349532 a(0) = a(1) = 1; a(n) = 4 * a(n-1) + Sum_{i=0..n-1} Sum_{j=0..n-i-1} a(i) * a(j) * a(n-i-j-1).
%F A349532 a(n) = Sum_{k=0..n} binomial(n-1,k-1) * binomial(3*k,k) * 4^(n-k) / (2*k+1).
%F A349532 a(n) = 4^(n-1)*F([4/3, 5/3, 1-n], [2, 5/2], -3^3/2^4), where F is the generalized hypergeometric function. - _Stefano Spezia_, Nov 21 2021
%F A349532 a(n) ~ 43^(n + 1/2) / (3 * sqrt(Pi) * n^(3/2) * 2^(2*n + 2)). - _Vaclav Kotesovec_, Nov 25 2021
%t A349532 nmax = 21; A[_] = 0; Do[A[x_] = 1 + x A[x]^3/(1 - 4 x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
%t A349532 a[0] = a[1] = 1; a[n_] := a[n] = 4 a[n - 1] + 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 A349532 Table[Sum[Binomial[n - 1, k - 1] Binomial[3 k, k] 4^(n - k)/(2 k + 1), {k, 0, n}], {n, 0, 21}]
%Y A349532 Cf. A001764, A025230, A307678, A349318, A349531.
%K A349532 nonn
%O A349532 0,3
%A A349532 _Ilya Gutkovskiy_, Nov 21 2021