Expected visits to the origin by a simple random walk
= Expected visits to the origin by a simple random walk
For the symmetric walk and $V_{2n}=|\{0\leq j\leq2n:S_j=0\}|$,
$$
\mathbb EV_{2n}=\sum_{k=0}^n\frac{\binom{2k}{k}}{4^k}.
$$
Since the central binomial coefficient satisfies $\binom{2k}{k}/4^k\asymp k^{-1/2}$, this expectation is of order $\sqrt n$.