A171213 G.f.: A(x) = x + x*A(A(3x)).

1, 3, 54, 3402, 618921, 318392208, 474852630879, 2094575471899362, 27570620677894020891, 1086589159409074932937593, 128377839432663886566934695744, 45490432595875817814676362194769627

Offset

1,2

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)).

Links

Table of n, a(n) for n=1..12.

Example

G.f.: A(x) = x + 3*x^2 + 54*x^3 + 3402*x^4 + 618921*x^5 +...
A(A(x)) = x + 6*x^2 + 126*x^3 + 7641*x^4 + 1310256*x^5 +...+ a(n)*x^n/3^(n-1) +...
As a formal series involving products of iterations of the g.f.,
A(x) = x + x*A(3x) + x*A(3x)*A(3A(3x) + x*A(3x)*A(3A(3x))*A(3A(3A(3x))) +...
which, upon replacing x with A(3x), yields:
A(A(3x)) = A(3x) + A(3x)*A(3A(3x)) + A(3x)*A(3A(3x))*A(3A(3A(3x))) +...
thus A(x) = x + x*A(A(3x)).

Prog

(PARI) {a(n, q=3)=local(A=x+x^2); for(i=1, n, A=x+x*subst(A, x, subst(A, x, q*x+O(x^n)))); polcoeff(A, n)}

Crossrefs

Cf. A171212 (q=2), A171214 (q=1/3).

Sequence in context: A224368 A091826 A091796 * A006025 A049414 A003027

Adjacent sequences: A171210 A171211 A171212 * A171214 A171215 A171216

Keyword

nonn

Author

Paul D. Hanna, Dec 08 2009

Status

approved