A171214 G.f. A(x) satisfies A(x) = x + x*A(A(x/3)) = Sum_{n>=1} a(n)*x^n/3^(n*(n+1)/2).

1, 1, 2, 10, 137, 5296, 588365, 190088818, 179954321171, 501722122937995, 4134242130461174144, 100943613343624534183723, 7317423203727305175501741434, 1577227642328692213664066391691150

Offset: 1

Author

Paul D. Hanna, Dec 08 2009

Keywords

• nonn

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?

Examples

			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...
		

Crossrefs

Cf. A171212 (q=2), A171213 (q=3), A030266 (q=1).

Programs

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)}