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%I #9 Apr 21 2024 03:05:00
%S 1,1,9,305,39705,20412737,41846783913,342892875489361,
%T 11236600170415809849,1472826135905484728387681,
%U 772188014962631262957890704329,1619397184353040716422147490531778929,13584491414647344530078887450781292845554521
%N G.f. A(x) satisfies: A(x) = 1 / (1 + x - 2 * x * A(4*x)).
%F a(0) = 1; a(n) = -a(n-1) + Sum_{k=0..n-1} 2^(2*k+1) * a(k) * a(n-k-1).
%F a(n) ~ c * 2^(n^2), where c = 2^(7/8) / EllipticTheta(2, 0, 1/sqrt(2)) = 0.6091497110662286155211146043057245512950999410185846745870491125003511... (same constant as in A165941). - _Vaclav Kotesovec_, Nov 03 2021, updated Apr 21 2024
%t nmax = 12; A[_] = 0; Do[A[x_] = 1/(1 + x - 2 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[2^(2 k + 1) a[k] a[n - k - 1], {k, 0, n - 1}]; Table[a[n], {n, 0, 12}]
%Y Cf. A001003, A015085, A165941, A348188, A348901.
%K nonn
%O 0,3
%A _Ilya Gutkovskiy_, Nov 03 2021