Inverse of series $z+a_2z^2+...$ Let $p$ be a power series with integer coefficients of the special form $p(z)=z+a_2z^2+a_3z^3+..$. I wonder if the inverse (composition not $1/p$) series has again integer coefficients. 
I have calculated some of such series so I guess yes. What do you think?

Given
$$
f(x)=x+a_2x^2+a_3x^3+\dots\tag{1}
$$
and
$$
g(x)=x+b_2x^2+b_3x^3+\dots\tag{2}
$$
formally, for $n\ge2$, the coefficient of $x^n$ in $g\circ f$ is
$$
0=a_n+b_n+\left[x^n\right]\sum_{k=2}^{n-1}b_k\left(x+a_2x^2+a_3x^3+\dots+a_{n-1}x^{n-1}\right)^k\tag{3}
$$
For $n=2$, $(3)$ says that $b_2=-a_2$. Then inductively, since $a_n\in\mathbb{Z}$, $(3)$ guarantees that $b_n\in\mathbb{Z}$.
Even if we only know for $2\le k\le n$, that $a_k\in\mathbb{Z}$, $(3)$ guarantees that for $2\le k\le n$, we have $b_k\in\mathbb{Z}$.

For example:
$b_2=-a_2$
$b_3=2a_2^2-a_3$
$b_4=5a_2a_3-5a_2^2-a_4$
$b_5=14a_2^4-21a_2^2a_3+3a_3^2+6a_2a_4-a_5$
