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%I #7 Oct 05 2024 17:07:41
%S 1,1,-3,-34,495,13631,-467404,-23984426,1490938299,123999435015,
%T -12164649041259,-1497474725212924,212746558833692052,
%U 36393896155519042476,-7062273474686464802160,-1603475573855830444120802,407344895625777134555939139,118552169162473363108837155199,-38177398083353809033748641523305
%N a(0) = 1; a(n) = Sum_{k=0..n-1} (-1)^k * (k+1)^2 * a(k) * a(n-k-1).
%F G.f. A(x) satisfies: A(x) = 1 / (1 - x * A(-x) + 3 * x^2 * A'(-x) - x^3 * A''(-x)).
%t a[0] = 1; a[n_] := a[n] = Sum[(-1)^k (k + 1)^2 a[k] a[n - k - 1], {k, 0, n - 1}]; Table[a[n], {n, 0, 18}]
%t nmax = 18; A[_] = 0; Do[A[x_] = 1/(1 - x A[-x] + 3 x^2 A'[-x] - x^3 A''[-x]) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
%Y Cf. A090192, A105523, A376095, A376134, A376135.
%K sign
%O 0,3
%A _Ilya Gutkovskiy_, Sep 11 2024