Is there any way to find closed form for the sum (where k is positive integer)
$$S = \sum_{i = 1}^{n}\sum_{j = 0}^{i} \left( \frac{j}{i} \right) ^ k$$
Using Faulhaber's formula I got
$$S = \frac{1}{k + 1} \sum_{r = 1}^{k + 1} (-1) ^ {\delta_{k, r}} \binom{k + 1}{r}B_{k + 1 - r} H_{n}^{(k - r)}$$
but I don't know how to continue.
