Composition coefficients of power series ($[x^n]f(g(x))={\sum }_{m=0}^n{f}_m[x^n]g(x{)}^m(g(0)=0)$)

For power series $f(z)={\sum }_{m\ge 0}{f}_m{z}^m$ and $g(x)={\sum }_{j\ge 1}{g}_j{x}^j$, their function composition is well defined as a formal power series: the coefficient of $x^n$ receives contributions only from $m\le n$, since $g$ has zero constant term. For $m\ge 1$, expand the coefficient asThe $m=0$ term is one for $n=0$ and zero otherwise. The first four coefficients of the function composition are therefore $f_0$, $f_1{g}_1$, $f_1{g}_2+f_2{g}_1^2$, and $f_1{g}_3+2f_2{g}_1{g}_2+f_3{g}_1^3$. For convergent power series, the same calculation holds wherever $g(x)$ lies inside the radius of convergence of $f$.