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A354733 a(0) = a(1) = 1; a(n) = 2 * Sum_{k=0..n-2} a(k) * a(n-k-2).

%I A354733 #15 Nov 05 2023 09:00:42
%S A354733 1,1,2,4,10,24,64,168,464,1280,3624,10304,29728,86240,252480,743040,
%T A354733 2200640,6547200,19571200,58727680,176883200,534476800,1619912320,
%U A354733 4923070464,14999764480,45807916544,140196076544,429931051008,1320905583616,4065358827520
%N A354733 a(0) = a(1) = 1; a(n) = 2 * Sum_{k=0..n-2} a(k) * a(n-k-2).
%H A354733 Seiichi Manyama, <a href="/A354733/b354733.txt">Table of n, a(n) for n = 0..1000</a>
%F A354733 G.f. A(x) satisfies: A(x) = 1 + x + 2 * (x * A(x))^2.
%F A354733 G.f.: (1 - sqrt(1 - 8 * x^2 * (1 + x))) / (4 * x^2).
%F A354733 a(n) ~ 5^(1/4) * (1 + sqrt(5))^(n+2) / (8 * sqrt(Pi) * n^(3/2)). - _Vaclav Kotesovec_, Jun 04 2022
%F A354733 a(n) = Sum_{k=0..floor(n/2)} 2^k * binomial(k+1,n-2*k) * A000108(k). - _Seiichi Manyama_, Nov 05 2023
%t A354733 a[0] = a[1] = 1; a[n_] := a[n] = 2 Sum[a[k] a[n - k - 2], {k, 0, n - 2}]; Table[a[n], {n, 0, 29}]
%t A354733 nmax = 29; CoefficientList[Series[(1 - Sqrt[1 - 8 x^2 (1 + x)])/(4 x^2), {x, 0, nmax}], x]
%o A354733 (PARI) a(n) = sum(k=0, n\2, 2^k*binomial(k+1, n-2*k)*binomial(2*k, k)/(k+1)); \\ _Seiichi Manyama_, Nov 05 2023
%Y A354733 Cf. A000108, A007477, A052701, A151374, A354734, A354735, A354736.
%K A354733 nonn
%O A354733 0,3
%A A354733 _Ilya Gutkovskiy_, Jun 04 2022