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%I #9 Nov 17 2021 04:54:20
%S 1,0,1,0,2,-1,5,-6,16,-28,62,-125,267,-565,1213,-2618,5686,-12418,
%T 27248,-60048,132848,-294930,656878,-1467257,3286219,-7378239,
%U 16603459,-37441989,84599855,-191501531,434224405,-986161958,2243009870,-5108859820,11651743902
%N G.f. A(x) satisfies: A(x) = 1 + x * A(x) / (1 - x) - x * A(x)^2.
%F a(0) = 1; a(n) = Sum_{k=0..n-1} a(k) * (1 - a(n-k-1)).
%F a(n) = 1 - Sum_{k=0..n-1} (-1)^k * A007477(k).
%F a(n) ~ 3^(1 + n) * (1/((1 - 2/(19 - 3*sqrt(33))^(1/3) - (1/2)*(19 - 3*sqrt(33))^(1/3))^n * ((19 - 3*sqrt(33))^(1/6)*(2 + (19 - 3*sqrt(33))^(1/3))^2 * n^(3/2) * sqrt(((-1951699 + 339747*sqrt(33))*Pi) / (-70717234 + 12310290*sqrt(33) + (19 - 3*sqrt(33))^(2/3) * (-3903398 + 679494*sqrt(33)) + (19 - 3*sqrt(33))^(1/3) * (-35358617 + 6155145*sqrt(33))))))). - _Vaclav Kotesovec_, Nov 17 2021
%t nmax = 34; A[_] = 0; Do[A[x_] = 1 + x A[x]/(1 - x) - x A[x]^2 + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
%t a[0] = 1; a[n_] := a[n] = Sum[a[k] (1 - a[n - k - 1]), {k, 0, n - 1}]; Table[a[n], {n, 0, 34}]
%Y Cf. A000108, A007477, A215973, A349014, A349016.
%K sign
%O 0,5
%A _Ilya Gutkovskiy_, Nov 05 2021
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