For all natural numbers $a$, is there a known closed form of $ \sum_{i=1}^{n-k} {n-1-i\choose k-1}i^a + \sum_{i=1}^k {n-1-i\choose n-1-k}$, where $k$ is fixed?

For example, letting $k=1$ gives the basic powersum of which the closed form is Faulhaber's formula for $a$. Is there a Faulhaber-like formula for all the rest of the possible fixed $k$ values?

I'm asking this because this corresponds to the values in Pascal triangles where the left diagonal is the sequence $m^a$ and the right diagonal is 1.