A171212 G.f.: A(x) = x + x*A(A(2x)).

1, 2, 16, 320, 12928, 985088, 140861440, 38451150848, 20403322617856, 21307854867660800, 44110759073910095872, 181739941085108158595072, 1493546441998961207249207296, 24512116566896662943648857456640

Offset

0,2

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=0..13.

Example

G.f.: A(x) = x + 2*x^2 + 16*x^3 + 320*x^4 + 12928*x^5 +...
A(A(x)) = x + 4*x^2 + 40*x^3 + 808*x^4 + 30784*x^5 + 2200960*x^6 +...+ a(n)*x^n/2^(n-1) +...
As a formal series involving products of iterations of the g.f.,
A(x) = x + x*A(2x) + x*A(2x)*A(2A(2x) + x*A(2x)*A(2A(2x))*A(2A(2A(2x))) +...
which, upon replacing x with A(2x), yields:
A(A(2x)) = A(2x) + A(2x)*A(2A(2x)) + A(2x)*A(2A(2x))*A(2A(2A(2x))) +...
thus A(x) = x + x*A(A(2x)).

Prog

(PARI) {a(n, q=2)=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. A171213 (q=3), A171214 (q=1/3).

Sequence in context: A282787 A137729 A278589 * A282392 A336522 A294039

Adjacent sequences: A171209 A171210 A171211 * A171213 A171214 A171215

Keyword

nonn

Author

Paul D. Hanna, Dec 08 2009

Status

approved