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a(0) = a(1) = 1; a(n) = a(n-1) + a(n-2) + Sum_{k=0..n-1} a(k) * a(n-k-1).
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%I #35 Jul 05 2020 23:36:25

%S 1,1,4,14,54,220,934,4090,18344,83850,389214,1829736,8693962,41685714,

%T 201442188,980091814,4797070022,23603701828,116688837886,579312087802,

%U 2887020896016,14437318756818,72424982972862,364366674463824,1837954750285458

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

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

%F G.f.: (1 - x - x^2 - sqrt(1 - 6*x + 3*x^2 + 2*x^3 + x^4)) / (2*x).

%t a[0] = a[1] = 1; a[n_] := a[n] = a[n - 1] + a[n - 2] + Sum[a[k] a[n - k - 1], {k, 0, n - 1}]; Table[a[n], {n, 0, 24}]

%t nmax = 24; CoefficientList[Series[(1 - x - x^2 - Sqrt[1 - 6 x + 3 x^2 + 2 x^3 + x^4])/(2 x), {x, 0, nmax}], x]

%Y Cf. A002212, A004148, A006318, A085139, A128720, A143330, A171416, A175934, A245734.

%K nonn

%O 0,3

%A _Ilya Gutkovskiy_, Jul 05 2020