OFFSET
1,3
COMMENTS
More generally, if F(x) = x + x*F(F(qx)), then
F(x) = x + x*F(qx) + x*F(qx)*F(qF(qx) + x*F(qx)*F(qF(qx))*F(qF(qF(qx))) +...
with a simple solution at q=1/2:
F(x) = x/(1-x/2) satisfies F(x) = x + x*F(F(x/2)).
At q=1, F(x,q=1) is the g.f. of A030266.
QUESTIONS regarding convergence of F(x,q) as a power series in x.
(1) What is Q, the maximum q below which a radius of convergence exists? Is Q=1?
(2) What is the radius of convergence for a given q < Q?
EXAMPLE
G.f.: A(x) = x + x^2/3 + 2*x^3/3^3 + 10*x^4/3^6 + 137*x^5/3^10 + 5296*x^6/3^15 +...+ a(n)*x^n/3^(n(n-1)/2) +...
A(x) = x + x*A(x/3) + x*A(x/3)*A(A(x/3)/3) + x*A(x/3)*A(A(x/3)/3)*A(A(A(x/3)/3)/3) +...
A(A(x)) = x + 2*x^2/3 + 10*x^3/3^3 + 137*x^4/3^6 + 5296*x^5/3^10 +...
SUMS OF SERIES at certain arguments.
A(1) = 1.423879975541542344910599787693637973194...
A(1/3) = 0.373293286580877833612329400906044642790...
A(A(1/3)) = A(1) - 1 = 0.42387997554...
A(A(1)) = 2.387414460111728675082753594461076041830...
A(3) = 3 + 3*A(A(1)) = 10.16224338033518602524826...
PROG
(PARI) {a(n)=local(A=x+x^2); for(i=1, n, A=x+x*subst(A, x, subst(A, x, x/3+O(x^n)))); 3^(n*(n-1)/2)*polcoeff(A, n)}
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Dec 08 2009
STATUS
approved