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%I #6 Nov 07 2021 02:36:55
%S 1,1,-5,-101,5293,869269,-420787937,-614362594985,2685998620138297,
%T 35251053957604379689,-1387622522805833315933693,
%U -163878220402091372424795125261,58060742480730955957157145945031525,61711834213019891772066352604323845604861
%N G.f. A(x) satisfies: A(x) = 1 / (1 + x - 2 * x * A(-3*x)).
%F a(0) = 1; a(n) = -a(n-1) + 2 * Sum_{k=0..n-1} (-3)^k * a(k) * a(n-k-1).
%t nmax = 13; A[_] = 0; Do[A[x_] = 1/(1 + x - 2 x 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] + 2 Sum[(-3)^k a[k] a[n - k - 1], {k, 0, n - 1}]; Table[a[n], {n, 0, 13}]
%Y Cf. A001003, A015098, A348188, A349038, A349046.
%K sign
%O 0,3
%A _Ilya Gutkovskiy_, Nov 06 2021