The simple random walk on the circle graph $\mathbb{Z}/N\mathbb{Z}$ of size $N$ can be realized as the class modulo $N$ of simple random walk on $\mathbb{Z}$, hence
$$
p_{00}^{(2n)}(\mathbb{Z}/N\mathbb{Z})=\sum_{k\in\mathbb{Z}}p_{0,kN}^{(2n)}(\mathbb{Z}).
$$
Now, for every $k$,
$$
p_{0,2k}^{(2n)}(\mathbb{Z})=2^{-2n}{2n\choose n+k},
$$
with the convention that ${2n\choose i}=0$ for every $i\le -1$ or $i\ge 2n+1$.