For Q.1 the answer 1/2 comes straight out of Faulhaber's formula. But that is independent of p, which makes it very tempting to think that the answer to Q.2 might be the same.
For Q.1 the answer 1/2 comes straight out of Faulhaber's formula. But that is independent of p, which makes it very tempting to think that the answer to Q.2 might be the same.