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%I #6 Nov 06 2021 20:15:46
%S 1,1,2,0,17,29,1459,-4233,1056763,11355763,6957281732,-209598234798,
%T 410408244241271,37950250148465939,218184080600974976674,
%U -60068553848055713514168,1043447774597599997266176403,864004926526955255880635472763,44918734284964096829849186288888390
%N G.f. A(x) satisfies: A(x) = 1 / (1 - x - x^2 * A(-3*x)).
%F a(0) = 1; a(n) = a(n-1) + Sum_{k=0..n-2} (-3)^k * a(k) * a(n-k-2).
%t nmax = 18; A[_] = 0; Do[A[x_] = 1/(1 - x - x^2 A[-3 x]) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
%t a[0] = 1; a[n_] := a[n] = a[n - 1] + Sum[(-3)^k a[k] a[n - k - 2], {k, 0, n - 2}]; Table[a[n], {n, 0, 18}]
%Y Cf. A001006, A015098, A348879, A349035, A349037.
%K sign
%O 0,3
%A _Ilya Gutkovskiy_, Nov 06 2021