Valuation of a near-central binomial coefficient
= Valuation of a near-central binomial coefficient
If $k\geq1$ and $k+2\leq p\leq2k+1$ is prime, then
$$
v_p\left(\binom{2k+1}{k+1}\right)=1.
$$
Indeed, neither $(k+1)!$ nor $k!$ contains a factor $p$, while $(2k+1)!$ contains $p$ exactly once because $2p>2k+1$.