Q:

Given a formal power series $f_1$, can we produce another formal power series $f_2$ suing computer program s.t.  $f_1 \circ f_2=f_2 \circ f_1$? 

When two formal power series commutes ? 
Algorithm:
Let $f = \sum a_n x^n$ and $g = \sum b_n x^n$ be two formal series without constant term.  Then
$$
f \circ g = \sum a_n g(x)^n = \sum a_n \left(\sum b_k x^k \right)^n
$$
Now, since
$$
\left(\sum b_k x^k \right)^n = \sum_{i_1 + \dotsm + i_k = n} b_{i_1} \dotsm b_{i_k} x^n
$$
one gets
$$
f \circ g = \sum a_n \left(\sum_{i_1 + \dotsm + i_k = n} b_{i_1} \dotsm b_{i_k}\right)x^n
$$
Therefore $f$ and $g$ commute in the sense of composition of functions if and only if, for all $n$,
$$
a_n \left(\sum_{i_1 + \dotsm + i_k = n} b_{i_1} \dotsm b_{i_k}\right) = b_n \left(\sum_{i_1 + \dotsm + i_k = n} a_{i_1} \dotsm a_{i_k}\right)
$$
This is the general criteria of commuting two formal power series. 

My question is about computer programming.  That is, suppose we are given a formal power series $f_1(x)$ without constant term, knowing its first $10$ terms or first $10$ coefficients of any finite number of coefficients.  Is it possible to write a computer programming like $PARI/GP$ $C++$, Matlab etc.  using above algorithm in order to get a formal power series $f_2(x)$ without constant term such that $f_1 \circ f_2=f_2 \circ f_1$ ?  At least, can we get the first $10$ terms of coefficients of $f_2(x)$?  

Actually I need some example of formal power series which commute each other in the sense of composition of functions.  

A:

You are aware that your formal series must start at $n=1$ for the composition to be well-defined ?  If $f \in xk^*+x^2 k[[x]]$ then why not take $ g=f^{-1}$ ?  Please clarify.