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%I #41 Jul 15 2020 07:30:19
%S 1,1,1,1,2,4,8,18,48,144,456,1560,5808,23184,98160,440832,2101824,
%T 10588608,56104128,312013440,1818498816,11082682368,70467474816,
%U 466680045312,3214497245184,22994283345408,170573216656896,1310482565462016,10415453732637696
%N a(0) = a(1) = a(2) = 1; a(n) = Sum_{k=0..n-3} binomial(n-3,k) * a(k) * a(n-k-3).
%C Shifts 3 places left when e.g.f. is squared.
%H Georg Fischer, <a href="/A333497/b333497.txt">Table of n, a(n) for n = 0..500</a>
%F E.g.f. A(x) satisfies: A(x) = 1 + x + x^2/2 + Integral( Integral( Integral A(x)^2 dx) dx) dx.
%t a[0] = a[1] = a[2] = 1; a[n_] := a[n] = Sum[Binomial[n - 3, k] a[k] a[n - k - 3], {k, 0, n - 3}]; Table[a[n], {n, 0, 28}]
%t nmax = 28; A[_] = 0; Do[A[x_] = 1 + x + x^2/2 + Integrate[Integrate[Integrate[A[x]^2, x], x], x] + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x] Range[0, nmax]!
%Y Cf. A000142, A007558, A307970, A336009, A336010.
%K nonn
%O 0,5
%A _Ilya Gutkovskiy_, Jul 04 2020