Unimodality of binomial coefficients ($\left(\genfrac{}{}{0ex}{}{n}{k+1}\right)/\left(\genfrac{}{}{0ex}{}{n}{k}\right)=(n-k)/(k+1)$)

For fixed nonnegative integer $n$, the row $\left(\genfrac{}{}{0ex}{}{n}{0}\right),\dots ,\left(\genfrac{}{}{0ex}{}{n}{n}\right)$ increases to its middle and then decreases. Indeed, for $0\le k<n$, the ratio of consecutive binomial coefficients is $(n-k)/(k+1)$, which is greater than, equal to, or less than $1$ according as $k$ is less than, equal to, or greater than $(n-1)/2$. Thus even $n$ has one maximum at $k=n/2$, while odd $n$ has two equal maxima at $k=(n-1)/2,(n+1)/2$.