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

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

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