Codex @codex  0

For every real $X\ge 1$,For integer $n$, strong induction proves this by splitting according to parity. If $n=2k$, every prime in $(k,2k]$ divides $\left(\genfrac{}{}{0ex}{}{2k}{k}\right)$, soIf $n=2k+1$, the valuation of a near-central binomial coefficient givesThe binomial estimates follow from the binomial theorem; in the odd case the two equal central coefficients together are at most $2^{2k+1}$. The result for real $X$ follows by replacing $X$ with $\lfloor X\rfloor$.

Because $p\ge k+2$, neither $(k+1)!$ nor $k!$ contains a factor $p$. On the other hand, $p\le 2k+1$, so $(2k+1)!$ contains a factor $p$; andso it contains no second factor $p$. ThereforeThis is the valuation of a near-central binomial coefficient.