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%I #9 Oct 31 2015 14:23:01
%S 1,1,2,3,5,9,10,22,20,51,40,114,67,230,130,474,203,891,380,1725,575,
%T 3108,1032,5718,1524,9986,2600,17568,3874,30048,6290,50988,9420,85647,
%U 14450,140796,22195,233095,32260,373536,50656,609804,69464,956368
%N G.f. satisfies: A(x) = A(x^2)^2 + x*A(x^2)^3.
%H Paul D. Hanna, <a href="/A174512/b174512.txt">Table of n, a(n), n=0..8200.</a>
%F A series quadrisection of A(x) equals 2*x^2*A(x^4)^5.
%e G.f.: A(x) = 1 + x + 2*x^2 + 3*x^3 + 5*x^4 + 9*x^5 + 10*x^6 +...
%e A(x)^2 = 1 + 2*x + 5*x^2 + 10*x^3 + 20*x^4 + 40*x^5 + 67*x^6 +...
%e A(x)^3 = 1 + 3*x + 9*x^2 + 22*x^3 + 51*x^4 + 114*x^5 + 230*x^6 +...
%e A(x)^4 = 1 + 4*x + 14*x^2 + 40*x^3 + 105*x^4 + 260*x^5 + 594*x^6 +..
%e A(x)^5 = 1 + 5*x + 20*x^2 + 65*x^3 + 190*x^4 + 516*x^5 + 1300*x^6 +...
%e A(x)^6 = 1 + 6*x + 27*x^2 + 98*x^3 + 315*x^4 + 930*x^5 + 2546*x^6 +...
%e where the series bisections of A(x)^2 are:
%e [A(x)^2 - A(-x)^2]/2 = 2*x*A(x^2)^5 and
%e [A(x)^2 + A(-x)^2]/2 = A(x^2)^4 + x^2*A(x^2)^6.
%e The series bisections of A(x)^3 are:
%e [A(x)^3 - A(-x)^3]/2 = 3*x*A(x^2)^7 + x^3*A(x^2)^9 and
%e [A(x)^3 + A(-x)^3]/2 = A(x^2)^6 + 3*x^2*A(x^2)^8.
%e The series bisections of A(x)^4 are:
%e [A(x)^4 - A(-x)^4]/2 = 4*x*A(x^2)^9 + 4*x^3*A(x^2)^11 and
%e [A(x)^4 + A(-x)^4]/2 = A(x^2)^8 + 6*x^2*A(x^2)^10 + x^4*A(x^2)^12.
%o (PARI) {a(n)=local(A=1+x);for(i=1,n,A=subst(A,x,x^2+x*O(x^n))^2+x*subst(A,x,x^2+x*O(x^n))^3);polcoeff(A,n)}
%o for(n=0,50,print1(a(n),", "))
%Y Cf. A174513.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Apr 20 2010
%E Edited by _Paul D. Hanna_, Apr 22 2010
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