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G.f. satisfies: A(x) = 1 + x*A(x)^4/A(-x).
2

%I #5 Mar 30 2012 18:37:11

%S 1,1,5,26,195,1303,11076,81910,740151,5782175,54176573,438029432,

%T 4203769940,34798104500,339699218160,2860590892318,28283147265023,

%U 241296800029199,2409437282086511,20767852798378330,209017295575667771

%N G.f. satisfies: A(x) = 1 + x*A(x)^4/A(-x).

%F G.f. satisfies: A(x) - x*A(x)^4 = 1 + x^2*[A(x)*A(-x)]^3.

%F G.f. satisfies:

%F _ A(x) = Sum_{n>=1} x^n * A(x)^(3*n)/A(-x)^n;

%F _ A(x) = exp( Sum_{n>=1} x^n/n * A(x)^(3*n)/A(-x)^n ). [From Paul D. Hanna, Sep 30 2011]

%e A bisection of g.f. A(x) equals a bisection of A(x)^4:

%e A(x) = 1 + x + 5*x^2 + 26*x^3 + 195*x^4 + 1303*x^5 + 11076*x^6 + 81910*x^7 +...

%e A(x)^4 = 1 + 4*x + 26*x^2 + 168*x^3 + 1303*x^4 + 9744*x^5 + 81910*x^6 +...

%e so that A(x) - x*A(x)^4 = 1 + x^2*[A(x)*A(-x)]^3, where

%e [A(x)*A(-x)]^3 = 1 + 27*x^2 + 1332*x^4 + 82791*x^6 + 5800329*x^8 +...

%e A(x)*A(-x) = 1 + 9*x^2 + 363*x^4 + 20820*x^6 + 1397511*x^8 +...

%e Related expressions.

%e A(x) = 1 + x*A(x)^3/A(-x) + x^2*A(x)^6/A(-x)^2 + x^3*A(x)^9/A(-x)^3 +...

%e log(A(x)) = x*A(x)^3/A(-x) + x^2/2*A(x)^6/A(-x)^2*x^2 + x^3/3*A(x)^9/A(-x)^3 +...

%o (PARI) {a(n)=local(A=1+x+x*O(x^n));for(i=0,n,A=1+x*A^4/subst(A,x,-x));polcoeff(A,n)}

%o (PARI) {a(n)=local(A=1+x+x*O(x^n));for(i=1,n,A=sum(m=0,n,x^m*A^(3*m)/subst(A^m,x,-x+x*O(x^n))));polcoeff(A,n)}

%o (PARI) {a(n)=local(A=1+x+x*O(x^n));for(i=1,n,A=exp(sum(m=1,n,A^(3*m)*subst(A^-m,x,-x)*x^m/m)+x*O(x^n)));polcoeff(A,n)}

%Y Cf. A143339, A143340.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Aug 09 2008