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a(n) = (-1)^n + 2 * Sum_{k=0..n-1} a(k) * a(n-k-1).
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%I #15 Aug 02 2023 11:45:11

%S 1,1,5,21,105,553,3053,17405,101713,606033,3667797,22485477,139340985,

%T 871429497,5492959293,34862161869,222592918689,1428814897825,

%U 9215016141989,59684122637237,388045493943049,2531696701375689,16569559364596365,108758426952823709

%N a(n) = (-1)^n + 2 * Sum_{k=0..n-1} a(k) * a(n-k-1).

%C Inverse binomial transform of A151374.

%H Seiichi Manyama, <a href="/A337168/b337168.txt">Table of n, a(n) for n = 0..1000</a>

%F G.f. A(x) satisfies: A(x) = 1 / (1 + x) + 2*x*A(x)^2.

%F G.f.: (1 - sqrt(1 - 8*x / (1 + x))) / (4*x).

%F E.g.f.: exp(3*x) * (BesselI(0,4*x) - BesselI(1,4*x)).

%F a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n,k) * 2^k * Catalan(k).

%F a(n) ~ 7^(n + 3/2) / (2^(9/2) * sqrt(Pi) * n^(3/2)). - _Vaclav Kotesovec_, Nov 13 2021

%t a[n_] := a[n] = (-1)^n + 2 Sum[a[k] a[n - k - 1], {k, 0, n - 1}]; Table[a[n], {n, 0, 23}]

%t Table[Sum[(-1)^(n - k) Binomial[n, k] 2^k CatalanNumber[k], {k, 0, n}], {n, 0, 23}]

%t Table[(-1)^n Hypergeometric2F1[1/2, -n, 2, 8], {n, 0, 23}]

%Y Cf. A000108, A005043, A052701, A151374, A162326, A337169.

%K nonn

%O 0,3

%A _Ilya Gutkovskiy_, Jan 28 2021