%I #7 Nov 02 2021 09:49:30

%S 1,2,7,44,481,9254,326395,21927776,2874607189,744650622170,

%T 383510575423471,393869218949592212,807827718206737362889,

%U 3311287802485779192925838,27136007596894473408507305443,444677773080105539125038867872456,14572535437424416878539776253365375549

%N G.f. A(x) satisfies: A(x) = 1 / ((1 - x) * (1 - x * A(2*x))).

%F a(n) = 1 + Sum_{k=0..n-1} 2^k * a(k) * a(n-k-1).

%F a(n) ~ c * 2^(n*(n-1)/2), where c = 10.96416094535958612421479005398505892527943513193882801485045169159164... - _Vaclav Kotesovec_, Nov 02 2021

%t nmax = 16; A[_] = 0; Do[A[x_] = 1/((1 - x) (1 - x A[2 x])) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]

%t a[n_] := a[n] = 1 + Sum[2^k a[k] a[n - k - 1], {k, 0, n - 1}]; Table[a[n], {n, 0, 16}]

%Y Cf. A007317, A015083, A348858, A348859.

%K nonn

%O 0,2

%A _Ilya Gutkovskiy_, Nov 02 2021