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%I #20 Feb 08 2022 05:18:28
%S 1,1,1,-1,1,-3,17,-85,385,-1767,8929,-50633,312705,-2036267,13794417,
%T -97295069,717808897,-5549714767,44868094145,-377741383697,
%U 3298933836033,-29813463964115,278462029910993,-2685972391332837,26733375327601281,-274247228584531767
%N G.f. A(x) satisfies: A(x) = 1 + x + x^2 * A(x/(1 + 2*x)) / (1 + 2*x).
%C Shifts 2 places left under 2nd-order inverse binomial transform.
%F a(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_] = 1 + 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] = 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. A007472, A009235, A010739, A051139, A351184, A351185, A351186, A351187.
%K sign
%O 0,6
%A _Ilya Gutkovskiy_, Feb 04 2022