A174513 - OEIS

%I #10 Oct 31 2015 14:22:45

%S 1,1,2,4,5,14,12,44,22,117,54,316,88,756,208,1836,317,4126,690,9216,

%T 1098,19906,2160,41876,3556,87448,6226,175832,11088,356368,17232,

%U 693356,32990,1365733,45402,2593576,94821,4971646,115464,9271456,263226

%N G.f. satisfies: A(x) = A(x^2)^2 + x*A(x^2)^4.

%H Paul D. Hanna, <a href="/A174513/b174513.txt">Table of n, a(n) for n = 0..4099</a>

%F A series quadrisection of A(x) equals 2*x^2*A(x^4)^6.

%e G.f.: A(x) = 1 + x + 2*x^2 + 4*x^3 + 5*x^4 + 14*x^5 + 12*x^6 +...

%e A(x)^2 = 1 + 2*x + 5*x^2 + 12*x^3 + 22*x^4 + 54*x^5 + 88*x^6 +...

%e A(x)^3 = 1 + 3*x + 9*x^2 + 25*x^3 + 57*x^4 + 144*x^5 + 299*x^6 +...

%e A(x)^4 = 1 + 4*x + 14*x^2 + 44*x^3 + 117*x^4 + 316*x^5 + 756*x^6 +...

%e A(x)^6 = 1 + 6*x + 27*x^2 + 104*x^3 + 345*x^4 + 1080*x^5 + 3113*x^6 +...

%e A(x)^8 = 1 + 8*x + 44*x^2 + 200*x^3 + 782*x^4 + 2800*x^5 + 9252*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)^6 and

%e [A(x)^2 + A(-x)^2]/2 = A(x^2)^4 + x^2*A(x^2)^8.

%e The series bisections of A(x)^3 are:

%e [A(x)^3 - A(-x)^3]/2 = 3*x*A(x^2)^8 + x^3*A(x^2)^12 and

%e [A(x)^3 + A(-x)^3]/2 = A(x^2)^6 + 3*x^2*A(x^2)^10.

%e The series bisections of A(x)^4 are:

%e [A(x)^4 - A(-x)^4]/2 = 4*x*A(x^2)^10 + 4*x^3*A(x^2)^14 and

%e [A(x)^4 + A(-x)^4]/2 = A(x^2)^8 + 6*x^2*A(x^2)^12 + x^4*A(x^2)^16.

%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))^4);polcoeff(A,n)}

%o for(n=0,30,print1(a(n),", "))

%Y Cf. A174512.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Apr 20 2010

%E Edited by _Paul D. Hanna_, Apr 22 2010