%I #6 Nov 06 2021 20:15:40

%S 1,1,2,1,9,6,165,97,10970,8617,2838793,1206206,2912348749,3338391105,

%T 11938619074866,-3485058191151,195607339607544393,505337929567029942,

%U 12820529140255160177781,-40595263531274884237983,3360756421633193695872693450

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

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

%t nmax = 20; A[_] = 0; Do[A[x_] = 1/(1 - x - x^2 A[-2 x]) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]

%t a[0] = 1; a[n_] := a[n] = a[n - 1] + Sum[(-2)^k a[k] a[n - k - 2], {k, 0, n - 2}]; Table[a[n], {n, 0, 20}]

%Y Cf. A001006, A015097, A348878, A349036, A349037.

%K sign

%O 0,3

%A _Ilya Gutkovskiy_, Nov 06 2021