I would like to approximate the following when $n \gg k$.

$\sum_{y = k + 1}^n \frac{\sum_{m = 0}^{k - 1} {y - 2 \choose m} (y - 1)}{\sum_{m = 0}^k {y - 1 \choose m}}.$

The formula can be re-written as

$\sum_{y = k + 1}^n \frac{(y - 1) + \sum_{m = 1}^{k - 1} {y - 1 \choose m + 1} (m + 1)}{\sum_{m = 0}^k {y - 1 \choose m}}.$

However since the partial sum of binomial coefficient does not closed form, I could not see any way to further simplify the formula. Any help is much appreciated. Thanks!