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%I #8 Nov 06 2021 09:07:04
%S 1,2,8,52,552,10208,350112,23159760,3012389984,777296223040,
%T 399542726439936,409933997609848192,840366306053838941952,
%U 3443822768422065940362240,28218687132517064788995222528,462391421142204650963524251763968,15152566983759983965941543133445666304
%N G.f. A(x) satisfies: A(x) = (1 + x * A(2*x)) / (1 - x * A(x)).
%F a(0) = 1; a(n) = 2^(n-1) * a(n-1) + Sum_{k=0..n-1} a(k) * a(n-k-1).
%F a(n) ~ c * 2^(n*(n-1)/2), where c = 11.40022022373995418911523299051117421707893086825818379118899572625286143... - _Vaclav Kotesovec_, Nov 06 2021
%t nmax = 16; A[_] = 0; Do[A[x_] = (1 + x A[2 x])/(1 - x A[x]) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
%t a[0] = 1; a[n_] := a[n] = 2^(n - 1) a[n - 1] + Sum[a[k] a[n - k - 1], {k, 0, n - 1}]; Table[a[n], {n, 0, 16}]
%Y Cf. A006318, A047891, A135867, A348875, A349013.
%K nonn
%O 0,2
%A _Ilya Gutkovskiy_, Nov 05 2021
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