Which is the maximum value of $k^2 \binom{n}{k}$, for $k$ and $n$ integers?

Question

Given an integer $n$, what is the maximum value that $k^2 \binom{n}{k}$ can take, for $k$ integer?

I've done the case of which $\binom{n}{k}$ is maximum, but for this one I don't see where to begin with. Any hint?

Answer 1

Hint: if $f(k) = k^2 {n \choose k}$, then $$ \frac{f(k+1)}{f(k)} = \frac{(k+1)(n-k)}{k^2} $$ When is this $> 1$ or $< 1$?