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A217925 G.f. A(x) satisfies A(x) = 1 + x*A(x)*A(x^2)^2.

%I A217925 #14 Mar 25 2015 15:23:03
%S A217925 1,1,1,3,5,10,19,40,77,155,306,610,1207,2400,4760,9456,18765,37257,
%T A217925 73955,146813,291434,578524,1148434,2279720,4525487,8983421,17832976,
%U A217925 35399824,70271944,139495472,276910976,549691232,1091185133,2166094309,4299884233,8535634803,16943967775
%N A217925 G.f. A(x) satisfies A(x) = 1 + x*A(x)*A(x^2)^2.
%C A217925 What does this sequence count?
%H A217925 Vaclav Kotesovec, <a href="/A217925/b217925.txt">Table of n, a(n) for n = 0..1000</a>
%F A217925 a(n) ~ c * d^n, where d = 1.985085392419660786124534041173530134614822710253953085885966352..., c = 0.322822740100478716884116064042886830242825005622702339543369128... . - _Vaclav Kotesovec_, Aug 10 2014
%F A217925 a(n) = T(4*n+1,1), where T(n,m) = Sum_{k=1..(n-m)/4} C(m+k-1,k)*T((n-m)/2,2*k). - _Vladimir Kruchinin_, Mar 25 2015
%o A217925 (PARI)
%o A217925 N=66;  R=O('x^N);  x='x+R;
%o A217925 F = 1 + x;
%o A217925 { for (k=1,N+1, F = 1 + x * F * subst(F,'x,'x^2)^2 + R; ); }
%o A217925 Vec(F+O('x^N))
%o A217925 (Maxima)
%o A217925 T(n,m):=if n=m then 1 else sum(binomial(m+k-1,k)*T((n-m)/2,2*k),k,1,(n-m)/4);
%o A217925 makelist(T(4*n+1,1),n,0,25); /* _Vladimir Kruchinin_, Mar 25 2015 */
%Y A217925 Cf. A000108 (A(x) = 1 + x*A(x)^2), A000621 (A(x) = 1 + x*A(x)*A(x^2)).
%Y A217925 Cf. A036675 (A(x) = 1 + x*A(x)^2*A(x^2)), A101913 (A(x) = 1 + x*A(x)*A(x^3); for abs. values).
%K A217925 nonn
%O A217925 0,4
%A A217925 _Joerg Arndt_, Oct 15 2012