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%I #12 Nov 13 2021 06:28:42
%S 1,1,4,29,348,7078,257400,17567085,2321150956,603642816242,
%T 311497277686152,320223587003352866,657101019781977963480,
%U 2694116441965648648689708,22080982977564915182409980400
%N G.f. satisfies: A(x) = 1 + x*A(x)^2*A(2x).
%H Vaclav Kotesovec, <a href="/A168602/b168602.txt">Table of n, a(n) for n = 0..80</a>
%F a(0) = 1; a(n) = Sum_{i=0..n-1} Sum_{j=0..n-i-1} 2^i * a(i) * a(j) * a(n-i-j-1). - _Ilya Gutkovskiy_, Nov 12 2021
%F a(n) ~ c * 2^(n*(n-1)/2)), where c = 8.922794036651587033803872123786584213844... - _Vaclav Kotesovec_, Nov 13 2021
%e G.f.: A(x) = 1 + x + 4*x^2 + 29*x^3 + 348*x^4 + 7078*x^5 +...
%e A(x)^2 = 1 + 2*x + 9*x^2 + 66*x^3 + 770*x^4 + 15084*x^5 +...
%t nmax = 20; A[_] = 0; Do[A[x_] = 1 + x*A[x]^2*A[2*x] + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x] (* _Vaclav Kotesovec_, Nov 13 2021 *)
%o (PARI) {a(n)=local(A=1+x+x*O(x^n)); for(i=0, n, A=1+x*A^2*subst(A, x, 2*x) ); polcoeff(A, n)}
%Y Cf. A168603.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Dec 05 2009
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