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a(n) = 1 + Sum_{k=0..n-3} binomial(n-2,k) * a(k).
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%I #5 May 21 2021 08:07:13

%S 1,1,1,2,4,8,20,57,171,548,1894,6998,27368,112653,486645,2201162,

%T 10397944,51161168,261571460,1386846249,7612315023,43190917004,

%U 252951090586,1527112817054,9492126182336,60677428545165,398489257136529,2686088269505042,18567557376240748

%N a(n) = 1 + Sum_{k=0..n-3} binomial(n-2,k) * a(k).

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

%t a[n_] := a[n] = 1 + Sum[Binomial[n - 2, k] a[k] , {k, 0, n - 3}]; Table[a[n], {n, 0, 28}]

%t nmax = 28; A[_] = 0; Do[A[x_] = (1 + x^2 A[x/(1 - x)])/((1 - x) (1 + x^2)) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]

%Y Cf. A000629, A210540, A344489, A344491, A344492, A344493.

%K nonn

%O 0,4

%A _Ilya Gutkovskiy_, May 21 2021