How to relate Molien's Theorem to this result. 
Let $G$ be a finite group and $(\pi, V )$ a finite dimensional $\mathbb{C}G$-module. For $n \ge 0$, let
$\mathbb{C}[V]_n$ be the space of homogeneous polynomial functions on $V$ of degree $n$. For a simple
$G$-representation $\rho$, denote by $a_n(\rho)$ the multiplicity of $\rho$ in $\mathbb{C}[V]_n$. Show that


$$
\sum_{n\ge 0} a_n(\rho) t^n = \sum_{g \in G} \frac{1}{|G|}\frac{\chi_\rho(g)}{\det(id_V-\pi(g)t)}.
$$

This seems to be very similar the the proof of Molien's Theorem, I just do not see how the multiplicity of the simple function gives a factor of $\chi_\rho(g)$ on each term of the sum on the right.
 A: The proof of Molien's theorem carries through for this generalization with no issues.
The degree $n$ homogeneous polynomial functions on a rep $V$ can be identified with the $n$th symmetric power of the dual rep, $\mathbb{C}[V]_n=\mathrm{Sym}^nV^{\ast}$. The only difference between the characters of $\mathrm{Sym}^nV$ and $\mathrm{Sym}^nV^{\ast}$ is that they are complex conjugates.
The multiplicity of an irrep $U$ within a rep $W$ is $\dim\hom_G(U,W)$, which is calculable as
$$ \dim\hom_G(U,W)=\dim(U^{\ast}\otimes W)^G=\frac{1}{|G|}\sum_{g\in G}\overline{\chi_U(g)}\chi_W(g). $$
Therefore, the multiplicities may be expressed as
$$ a_n(U)=\dim\hom_G(U,\mathrm{Sym}^nV^{\ast})=\frac{1}{|G|}\sum_{g\in G}\overline{\chi_U(g)}\, \overline{\chi_{\mathrm{Sym}^nV}(g)}. $$
Since this expression is real, we may discard the complex conjugations.
If $d=\dim V$ and the eigenvalues of $\rho_V(g)$ are $\lambda_1,\cdots,\lambda_d$ (imagine these are functions of $g$) then the eigenvalues of $\rho_{\mathrm{Sym}^nV}(g)$ are $\lambda_1^{k_1}\cdots\lambda_d^{k_d}$ with $k_1+\cdots+k_d=n$ (all counted with multiplicity).
Therefore we may write the function as
$$ \begin{array}{ll} M(t) & \displaystyle = \sum_{n=0}^\infty a_n(U)t^n \\[5pt]
& \displaystyle =\sum_{n=0}^\infty t^n\cdot\frac{1}{|G|}\sum_{g\in G} \chi_U(g)\chi_{\mathrm{Sym}^nV}(g) \\[5pt]
& \displaystyle =\frac{1}{|G|}\sum_{n=0}^\infty t^n\sum_{g\in G}\chi_U(g)\sum_{k_1+\cdots+k_d=n}\lambda_1^{k_1}\cdots\lambda_d^{k_d} \\[5pt]
& \displaystyle =\frac{1}{|G|}\sum_{g\in G}\chi_U(g)\sum_{k_1=0}^\infty\cdots\sum_{k_d=0}^\infty (\lambda_1t)^{k_1}\cdots(\lambda_dt)^{k_d} \\[5pt]
& \displaystyle = \frac{1}{|G|}\sum_{g\in G}\frac{\chi_U(g)}{(1-\lambda_1t)\cdots(1-\lambda_dt)} \\[5pt]
& \displaystyle = \frac{1}{|G|}\sum_{g\in G}\frac{\chi_U(g)}{\det(I_V-\rho_V(g)t)}. \end{array} $$
