OFFSET
1,5
COMMENTS
Compare the g.f. to the following property of G(x) = x*sqrt(1+4*x):
G(G(x))^2 = x^2 + 4*x^3 + 4*G(x)^3.
FORMULA
G.f. satisfies: A(x)^2 = A(-x)^2 + 4*x^3.
G.f. satisfies: A(-A(-x)) = x.
G.f. satisfies: A(x) = B(x) + x^3/B(x) where B(x) = (A(x) - A(-x))/2.
EXAMPLE
G.f.: A(x) = x + x^2 + x^3 - x^4 - 5*x^5 + 6*x^6 + 57*x^7 - 68*x^8 +...
Note that x^3 is the only odd power of x in A(x)^2:
A(x)^2 = x^2 + 2*x^3 + 3*x^4 - 11*x^6 + 117*x^8 - 2001*x^10 +...
Illustrate A(A(x))^2 = x^2 + 4*A(x)^3 by the expansions:
A(A(x))^2 = x^2 + 4*x^3 + 12*x^4 + 24*x^5 + 16*x^6 - 60*x^7 - 72*x^8 + 640*x^9 + 768*x^10 - 11160*x^11 - 12916*x^12 +...
A(x)^3 = x^3 + 3*x^4 + 6*x^5 + 4*x^6 - 15*x^7 - 18*x^8 + 160*x^9 + 192*x^10 - 2790*x^11 - 3229*x^12 +...
G.f. of odd bisection B(x) = (A(x) - A(-x))/2 begins:
B(x) = x + x^3 - 5*x^5 + 57*x^7 - 996*x^9 + 23487*x^11 +...
where A(x) = B(x) + x^3/B(x).
PROG
(PARI) {a(n)=local(A=x+x^2+x*O(x^n)); for(i=1, n, A=A-(subst(A, x, A)-x*sqrt(1+4*A^3/x^2))/2); polcoeff(A, n)}
CROSSREFS
KEYWORD
sign
AUTHOR
Paul D. Hanna, Jun 06 2011
STATUS
approved