A217925 - OEIS

%I #14 Mar 25 2015 15:23:03

%S 1,1,1,3,5,10,19,40,77,155,306,610,1207,2400,4760,9456,18765,37257,

%T 73955,146813,291434,578524,1148434,2279720,4525487,8983421,17832976,

%U 35399824,70271944,139495472,276910976,549691232,1091185133,2166094309,4299884233,8535634803,16943967775

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

%C What does this sequence count?

%H Vaclav Kotesovec, <a href="/A217925/b217925.txt">Table of n, a(n) for n = 0..1000</a>

%F a(n) ~ c * d^n, where d = 1.985085392419660786124534041173530134614822710253953085885966352..., c = 0.322822740100478716884116064042886830242825005622702339543369128... . - _Vaclav Kotesovec_, Aug 10 2014

%F 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 (PARI)

%o N=66;  R=O('x^N);  x='x+R;

%o F = 1 + x;

%o { for (k=1,N+1, F = 1 + x * F * subst(F,'x,'x^2)^2 + R; ); }

%o Vec(F+O('x^N))

%o (Maxima)

%o 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 makelist(T(4*n+1,1),n,0,25); /* _Vladimir Kruchinin_, Mar 25 2015 */

%Y Cf. A000108 (A(x) = 1 + x*A(x)^2), A000621 (A(x) = 1 + x*A(x)*A(x^2)).

%Y 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 nonn

%O 0,4

%A _Joerg Arndt_, Oct 15 2012