= Unimodality of binomial coefficients
{title2=$\binom n{k+1}/\binom nk=(n-k)/(k+1)$}
For fixed nonnegative <integer> $n$, the row $\binom n0,\ldots,\binom nn$ 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 $k=(n-1)/2,(n+1)/2$.
Back to article page
