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A monomial of weighted degree $n$ is $x^{n-2j}{y}^j$ with $0\le j\le \lfloor n/2\rfloor$. ThereforeThis is a quasipolynomial of period two, not eventually one polynomial: it equals $n/2+1$ for even $n$ and $(n+1)/2$ for odd $n$. The usual eventual-polynomial theorem for a graded algebra assumes a standard graded algebra, generated in degree one. Here the generator $y$ has degree two, so there is no contradiction.

A sufficient condition is that $A$ be a standard graded algebra: it is generated as a $k$-algebra by finitely many elements of degree one. The Hilbert-Serre theorem then makes $P_A(t)$ a rational function whose denominator, after cancellation, is a power of $1-t$. When the eventual Hilbert polynomial is nonzero,where the right side uses the order of the pole at $t=1$.