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G.f. A(x) satisfies: A(x) = x + x^2 * A(x/(1 - 2*x)) / (1 - 2*x).
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%I #14 Feb 04 2022 11:23:09

%S 0,1,0,1,4,13,44,173,792,4009,21608,122761,737340,4696341,31665076,

%T 224846037,1672266352,12976252561,104816144656,880061135057,

%U 7670326372532,69286959112797,647568753568636,6251768635591613,62255057942504968,638658964709824185

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

%C Shifts 2 places left under 2nd-order binomial transform.

%F a(0) = 0, a(1) = 1; a(n) = Sum_{k=0..n-2} binomial(n-2,k) * 2^k * a(n-k-2).

%t nmax = 25; A[_] = 0; Do[A[x_] = x + x^2 A[x/(1 - 2 x)]/(1 - 2 x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]

%t a[0] = 0; a[1] = 1; a[n_] := a[n] = Sum[Binomial[n - 2, k] 2^k a[n - k - 2], {k, 0, n - 2}]; Table[a[n], {n, 0, 25}]

%Y Cf. A000995, A004211, A007472, A351053, A351128, A351132, A351143, A351161.

%K nonn

%O 0,5

%A _Ilya Gutkovskiy_, Feb 03 2022