More Dice

We have $n^2$ dice labelled from $d_{1}$ to $d_{n^2}$ , where every dice has the numbers $1,2,3,...,n^2$ on its faces. We roll all the dice simultaneously, and denote $f(d_{i})$ as the number which comes up on $d_{i}$ . It is given that $d_{i} \ne d_{j}$ for all $i \ne j$ . Define: $g(n) = \displaystyle\sum_{k=1}^{\lfloor \dfrac{n^2}{2} \rfloor} f(d_{k})$ Given that $n$ is the smallest positive integer such that it is possible that $\displaystyle\sum_{k=1}^{\lfloor \dfrac{n^2}{2} \rfloor} f(d_{k}) : \displaystyle\sum_{k=\lfloor \dfrac{n^2}{2} \rfloor+1}^{n^2}f(d_{k}) = 2 : 5$ What is $g(n)$ ?