I know how to compute Hilbert series for the ordinary polynomial ring in $n$ variables. Consider now the following ring
$$ \mathbb{F}[x,y,z]/(xy^{3}) $$ where $(xy^{3})$ denote the ideal generated by $xy^{3}$ with the ordinary graduading, i.e. $deg(x) = deg(y) = deg(z) = 1$. Now for an $\mathbb{N}$-graded ring $R$ = $\bigoplus\limits_{i\geq0}R_{i}$ over $\mathbb{F}$ where $R_{i}$ denotes the i-th homogeneous components of $R$, the hilbert series is defined as follows
$$ \text{Hilb}(R;t) := \sum_{i\geq0}\text{dim}_{\mathbb{F}}R_{i} \cdot t^{i}. $$
My question is now, how the hilbert series does look like for the Special example above. I have to compute the Dimension of the $i$-th homogeneous components of $\mathbb{F}[x,y,z]/(xy^{3})$. Hope someone can help me