250 Follower Problem

Let the roots of the equation $x^m - x^2 - x - 250m = 0$ be $\omega_{1,m}, \omega_{2,m}, \ldots, \omega_{m,m}$ .

Find the value of

$\sum_{n=6}^{250} \sum_{k=1}^n {\omega^{n-2}_{k,n} }$

Inspiration