Let $f=\sum\limits_{n\geq 0}{f_n x^n}$ and $g=\sum\limits_{n\geq 1}{g_n x^n}$ be formal power series. The $x^n$ coefficient of $f(g)$ is $$ \sum\limits_{\mathbb{i} \in \mathcal{C}_{n}} {f_k \,g_{i_1}\cdots g_{i_k}} , $$ where $\mathcal{C}_{n} = \{ (i_1, \ldots, i_k) : 1 \leq k \leq n, i_1 + \cdots + i_k = n \}$ is the set of compositions of $n$ into $k$ parts.
Consider the power series $h=\sum\limits_{n\geq 0}{h_n x^n}$ with coefficient
$$ h_n = \sum\limits_{\mathbb{i} \in \mathcal{C}_{n}} {\binom{n}{k} f_k \,g_{i_1}\cdots g_{i_k}} . $$ Is there a "nice" way to express $h$ in terms of $f$ and $g$ or related generating series (eg. their e.g.f.s), where by "nice", I mean a formula that can be evaluated if $f$ and $g$ are at least rational functions with $g(0)=0$?
UPDATE
If we let $F=\sum\limits_{n\geq 0}{f_n \frac{x^n}{n!}}$ and $H=\sum\limits_{n\geq 0}{h_n \frac{x^n}{n!}}$ then, using the inverse Laplace transform $\mathcal{L}^{-1}$, we can write
$$H = \mathcal{L}^{-1}\{ F(s \,g(x/s))/s\}(1).$$
This is because $$\mathcal{L}^{-1}\{ s^{k-n}/s\}(1)=\frac{1}{(n-k)!}.$$
However this isn't what I want because
I want $h$, not $H$, and;
this formula is not "nice", as if $f=\frac{1}{1-x}$ (so $F=e^x$) and $g=\frac{x}{1-x}$, the expression for $H$ is an intractable integral, whereas $h=\frac{1}{2} + \frac{1}{2\sqrt{1-4x}}$.