This answer is essentially the same as that of Phil Tosteson, written before I saw that post. I also mention a non-Cohen-Macaulay example at the end.

If $R$ is Cohen-Macaulay (but not necessarily generated in degree one), then $R$ has associated with it a canonical module $\Omega(R)$ which can be graded so its Hilbert function agrees with $(-1)^d p_R(-n)$ for $n$ sufficiently large, where $d$ is the Krull dimension of $R$ or $\Omega(R)$. If $R$ has $n$ generators then it can be regarded as a module over the polynomial ring $k[x_1,\dots,x_n]$. One can then define $\Omega(R) = \mathrm{Ext}^{n-d}_A(R,A)$. If $R$ is not Cohen-Macaulay, then there are "correction terms" to the formula $p_R(-n) = (-1)^d\mathrm{HQ}(\Omega(R))$, where $\mathrm{HQ}$ denotes Hilbert quasipolynomial. Namely, $$ p_R(-n) = (-1)^d \sum_{i=0}^d (-1)^i\mathrm{HQ}\left( \mathrm{Ext}^{n-d+i}_A(R,A)\right). $$ (If $R$ is Cohen-Macaulay, then only the term indexed by $i=0$ doesn't vanish.) I don't know where this result is stated in precisely this form, but it is equivalent to Theorem 6.4 of my book Combinatorics and Commutative Algebra, second ed. Theorem 8.2 gives an example, stated in terms of Hilbert series rather than Hilbert quasipolynomials.

This answer is essentially the same as that of Phil Tosteson, written before I saw that post. I also mention a non-Cohen-Macaulay example at the end.

If $R$ is Cohen-Macaulay (but not necessarily generated in degree one), then $R$ has associated with it a canonical module $\Omega(R)$ which can be graded so its Hilbert function agrees with $(-1)^d p_R(-n)$ for $n$ sufficiently large, where $d$ is the Krull dimension of $R$ or $\Omega(R)$. If $R$ has $n$ generators then it can be regarded as a module over the polynomial ring $A=k[x_1,\dots,x_n]$. One can then define $\Omega(R) = \mathrm{Ext}^{n-d}_A(R,A)$. If $R$ is not Cohen-Macaulay, then there are "correction terms" to the formula $p_R(-n) = (-1)^d\mathrm{HQ}(\Omega(R))$, where $\mathrm{HQ}$ denotes Hilbert quasipolynomial. Namely, $$ p_R(-n) = (-1)^d \sum_{i=0}^d (-1)^i\mathrm{HQ}\left( \mathrm{Ext}^{n-d+i}_A(R,A)\right). $$ (If $R$ is Cohen-Macaulay, then only the term indexed by $i=0$ doesn't vanish.) I don't know where this result is stated in precisely this form, but it is equivalent to Theorem 6.4 of my book Combinatorics and Commutative Algebra, second ed. Theorem 8.2 gives an example, stated in terms of Hilbert series rather than Hilbert quasipolynomials.