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%I #5 Jul 05 2020 00:01:02
%S 1,1,1,1,1,2,4,8,16,34,82,226,678,2108,6892,23948,88532,344816,
%T 1401200,5925000,26146360,120743496,582606552,2926675112,15259183112,
%U 82458502624,461577781968,2674216518016,16013654472352,98968416103968,630595248710144
%N a(0) = ... = a(3) = 1; a(n) = Sum_{k=0..n-4} binomial(n-4,k) * a(k) * a(n-k-4).
%C Shifts 4 places left when e.g.f. is squared.
%F E.g.f. A(x) satisfies: A(x) = 1 + x + x^2/2 + x^3/6 + Integral( Integral( Integral( Integral A(x)^2 dx) dx) dx) dx.
%t a[0] = a[1] = a[2] = a[3] = 1; a[n_] := a[n] = Sum[Binomial[n - 4, k] a[k] a[n - k - 4], {k, 0, n - 4}]; Table[a[n], {n, 0, 30}]
%t nmax = 30; A[_] = 0; Do[A[x_] = 1 + x + x^2/2 + x^3/6 + Integrate[Integrate[Integrate[Integrate[A[x]^2, x], x], x], x] + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x] Range[0, nmax]!
%Y Cf. A000142, A007558, A307971, A333497, A336010.
%K nonn
%O 0,6
%A _Ilya Gutkovskiy_, Jul 04 2020
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