Inverse super-composition

[Xorter]

For a fixed point of zero, with a fixed point multiplier of 2, the general solution for the Abel function generated at the fixed point of zero is:
\( f(z) = 2x + \sum_{n=2}^{\infty}a_n z^n \)

\( \alpha(z) = \log_2(S(z))\;\;\; \) This is the Abel function for f(z) \( \;\alpha(f(z)) = \alpha(z)+1 \)

where S(z) is the formal Schröder equation solution;
\( S(f(z)) = 2\cdot S(z)\;\;\; S(z)=z+\sum_{n=2}^{\infty}b_n z^n\;\; \)
This is sometimes called Koenig's solution. It can be modified to work with any fixed point multiplier of k, |k|<>1. Using pari-gp one can easily write a program to generate the formal power series for S(x) given f(x).

I almost undestand it.
Okey, than what is the a[n] and b[n] in the sum formula?
I feel we are closer then ever before.
Could you show me this way with another example, please?
For instance, let us invastigate it: \( cos ^o ^N (x) = sin(x) \). (And cos' fixed point is ~0.739.) The question is that what N is and how I can calculate it.