Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%I #15 Aug 01 2023 09:33:42
%S 1,3,13,74,499,3719,29494,243888,2078431,18122369,160885449,
%T 1449268478,13213370392,121696581804,1130565483472,10581614352704,
%U 99685591788687,944490400760597,8994266558594671,86040075341770806,826423263373253923,7967095415955791687
%N G.f. A(x) satisfies A(x) = 1 / ((1 - 2 * x) * (1 - x * A(x)^2)).
%H Seiichi Manyama, <a href="/A349533/b349533.txt">Table of n, a(n) for n = 0..989</a>
%F 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 a(n) = Sum_{k=0..n} binomial(n+k,2*k) * binomial(3*k,k) * 2^(n-k) / (2*k+1).
%F 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 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 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 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 Table[Sum[Binomial[n + k, 2 k] Binomial[3 k, k] 2^(n - k)/(2 k + 1), {k, 0, n}], {n, 0, 21}]
%Y Cf. A001764, A064613, A199475, A349253, A349534, A349535.
%K nonn
%O 0,2
%A _Ilya Gutkovskiy_, Nov 21 2021