Let $Q$ be a polynomial of degree $k$ such that $$Q(n) = {k + 1 \choose n }^{-1}$$ for $n=0, 1, \ldots k$. Find $Q(n + 1)$.
I have been working with interpolating polynomials and other numerical methods but I have no idea where to begin solving this problem? If anyone has any tips, they are highly appreciated!