Suppose $R=k[x,y]/(f)$, where $k$ a field, $f=y^d+g\in k[x,y]$ and $g\in(x,y)^{d+1}$ with $d$ a fixed positive integer. The ideal $I=(x,y)$ is maximal in $R$.
The associated graded ring $A=G_I(R)$ should be $\bigoplus_{n=0}^{\infty}(x,y)^n/(x,y)^{n+1}$ by definition, viewed as an A-module. The ring $A$ is graded because $I^n/I^{n+1}\cdot I^m/I^{m+1}=I^{n+m}/I^{n+m+1}$.
If we let $\lambda=\dim_k$, how can we compute the Poincaré/Hilbert series $P_{\lambda}(A,t)=\Sigma_{n=0}^{\infty}\lambda(A_n)t^n$?
For every $A_n$ (which I think is equal to $I^n/I^{n+1}$) we need to compute the dimension over $k$ ($\dim=2$?). I think these dimensions should depend on $d$, but I don't see how to continue.
I'm absolutely clueless. Does anyone have a good hint to help me further? I think I'm overthinking this completely.
EDIT: I thought of something. We know that $g\in A_{d+1}$, the set of polynomials in $A$ of degree $d+1$. So $f\in A_d \cup A_{d+1}$. It's so hard to see for me how to get the Poincaré/Hilbert series of the associated graded ring in stead of just for $R$.