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a(n) = 1 + Sum_{k=0..n-2} binomial(n-2,k) * a(k).
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%I #8 Jan 29 2021 10:29:33

%S 1,1,2,3,6,14,36,101,308,1013,3562,13300,52482,218045,950614,4335563,

%T 20628882,102153978,525383324,2801105889,15455435864,88117352141,

%U 518391612686,3142762585120,19611454375090,125829007917417,829254498014570,5608225148263459

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

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

%t a[n_] := a[n] = 1 + Sum[Binomial[n - 2, k] a[k], {k, 0, n - 2}]; Table[a[n], {n, 0, 27}]

%Y Cf. A000994, A007476, A186021.

%K nonn

%O 0,3

%A _Ilya Gutkovskiy_, Jan 29 2021