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%I #5 Nov 06 2021 20:15:35
%S 1,2,-4,-88,5360,1395104,-1423111744,-5834786588032,95573832673124096,
%T 6263909110244685920768,-1642021136070472933898232832,
%U -1721790522986063937046243536001024,7221705990593287793620261453916626546688,121160150179535955805047509278599956409746825216
%N G.f. A(x) satisfies: A(x) = 1 / (1 - x - x * A(-4*x)).
%F a(0) = 1; a(n) = a(n-1) + Sum_{k=0..n-1} (-4)^k * a(k) * a(n-k-1).
%t nmax = 13; A[_] = 0; Do[A[x_] = 1/(1 - x - x A[-4 x]) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
%t a[0] = 1; a[n_] := a[n] = a[n - 1] + Sum[(-4)^k a[k] a[n - k - 1], {k, 0, n - 1}]; Table[a[n], {n, 0, 13}]
%Y Cf. A006318, A015099, A348877, A349032, A349033.
%K sign
%O 0,2
%A _Ilya Gutkovskiy_, Nov 06 2021