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A337169 a(n) = (-1)^n + 3 * Sum_{k=0..n-1} a(k) * a(n-k-1).

%I A337169 #13 Aug 02 2023 11:45:20
%S A337169 1,2,13,89,691,5720,49555,443630,4071595,38105342,362271823,
%T A337169 3488988101,33967656469,333752559392,3305347855573,32960499084305,
%U A337169 330664662067795,3335002912108670,33796042027030855,343940115478559699,3513702627928096681,36021007341027948032
%N A337169 a(n) = (-1)^n + 3 * Sum_{k=0..n-1} a(k) * a(n-k-1).
%C A337169 Inverse binomial transform of A005159.
%H A337169 Seiichi Manyama, <a href="/A337169/b337169.txt">Table of n, a(n) for n = 0..964</a>
%F A337169 G.f. A(x) satisfies: A(x) = 1 / (1 + x) + 3*x*A(x)^2.
%F A337169 G.f.: (1 - sqrt(1 - 12*x / (1 + x))) / (6*x).
%F A337169 E.g.f.: exp(5*x) * (BesselI(0,6*x) - BesselI(1,6*x)).
%F A337169 a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n,k) * 3^k * Catalan(k).
%F A337169 a(n) ~ 11^(n + 3/2) / (8 * 3^(3/2) * sqrt(Pi) * n^(3/2)). - _Vaclav Kotesovec_, Nov 13 2021
%t A337169 a[n_] := a[n] = (-1)^n + 3 Sum[a[k] a[n - k - 1], {k, 0, n - 1}]; Table[a[n], {n, 0, 21}]
%t A337169 Table[Sum[(-1)^(n - k) Binomial[n, k] 3^k CatalanNumber[k], {k, 0, n}], {n, 0, 21}]
%t A337169 Table[(-1)^n Hypergeometric2F1[1/2, -n, 2, 12], {n, 0, 21}]
%Y A337169 Cf. A000108, A005043, A005159, A337167, A337168.
%K A337169 nonn
%O A337169 0,2
%A A337169 _Ilya Gutkovskiy_, Jan 28 2021