Let's assume that we have two functions $f,g: \mathbb{R} \rightarrow \mathbb{R}$ of class $C^\infty$ such that $f(x)=g(x^p)$ for $x \in \mathbb{R}$, where $p \in \mathbb{N}$ is fixed.
I wish to calculate $g^{(n})(0)$ for $n \in \mathbb{N}$.
In particular case when $f(x)=\sum_{n=0}^\infty c_n x^n$, $f(x)=\sum_{n=0}^\infty d_n x^n$ for $x \in \mathbb{R}$, from $f(x)=g(x^p)$ for $x \in \mathbb{R}$, we receive that $c_n=0$ when $n$ is not of the form $n=kp$, where $k\in \mathbb{N}\cup\{0\}$. Hence $f(x)=\sum_{n=0}^\infty c_{pn}x^{pn}=f(x)=g(x^p)=\sum_{n=0}^\infty d_n x^{pn} \textrm{ for } x\in \mathbb{R},$
$\frac{g^{(n)}(0)}{n!}=d_n=c_{pn}=\frac{f^{(pn)}(0)}{(pn)!} \textrm{ for } n \in \mathbb{N} \cup \{0\}.$
It suggests that $g^{(n)}(0)=\frac{n!}{(np)!} f^{(np)}(0) \textrm{ for } n\in \mathbb{N}\cup \{0\}.$
I don't know how to prove it in general case, when $f,g$ are not analytic.