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A349331 G.f. A(x) satisfies A(x) = 1 + x * A(x)^4 / (1 - x).

%I A349331 #21 Nov 23 2024 09:09:01
%S A349331 1,1,5,31,219,1678,13570,114014,985542,8708099,78298727,714105907,
%T A349331 6590200215,61427125994,577456943614,5468604044500,52122539760992,
%U A349331 499613409224137,4813105582181533,46576519080852235,452545041339982871,4413071971740021275,43177663974461532959
%N A349331 G.f. A(x) satisfies A(x) = 1 + x * A(x)^4 / (1 - x).
%H A349331 Seiichi Manyama, <a href="/A349331/b349331.txt">Table of n, a(n) for n = 0..985</a>
%F A349331 a(n) = Sum_{k=0..n} binomial(n-1,k-1) * binomial(4*k,k) / (3*k+1).
%F A349331 a(n) ~ 283^(n + 1/2) / (2^(7/2) * sqrt(Pi) * n^(3/2) * 3^(3*n + 3/2)). - _Vaclav Kotesovec_, Nov 15 2021
%p A349331 a:= n-> coeff(series(RootOf(1+x*A^4/(1-x)-A, A), x, n+1), x, n):
%p A349331 seq(a(n), n=0..22);  # _Alois P. Heinz_, Nov 15 2021
%t A349331 nmax = 22; A[_] = 0; Do[A[x_] = 1 + x A[x]^4/(1 - x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
%t A349331 Table[Sum[Binomial[n - 1, k - 1] Binomial[4 k, k]/(3 k + 1), {k, 0, n}], {n, 0, 22}]
%o A349331 (PARI) {a(n) = my(A=[1]); for(m=1, n, A=concat(A, 0);
%o A349331 A[#A] = 1 + sum(k=1, m-1, (polcoeff(Ser(A)^4, k)) )); A[n+1]}
%o A349331 for(n=0, 30, print1(a(n), ", ")) \\ _Vaclav Kotesovec_, Nov 23 2024, after _Paul D. Hanna_
%Y A349331 Cf. A002212, A002293, A307678, A317133, A346646 (partial sums), A349332, A349333, A349334, A349335.
%K A349331 nonn
%O A349331 0,3
%A A349331 _Ilya Gutkovskiy_, Nov 15 2021