Concerns about the arithmetic genus of projective hypersurfaces led me to make the following combinatorial conjecture: ${d-1\choose n+1} =\sum_{i=0}^{n+1} (-1)^{n+i+1} {d\choose i}$ for $d \geq 1$, $n \geq 0$. If I made no mistakes in my code, I was also able to find some reasonably strong numerical evidence that this is, in fact, an identity. Unfortunately, my skill at combinatorics is sufficiently poor that while I might be able to prove the statement with a fair amount of effort, I doubt I could find an enlightening proof.
Can someone supply an enlightening proof of the statement above?
Obviously, a counterexample would also suffice, but I doubt there is one.
Additional note: I did not see a reasonable way to search for this specific identity, so it is quite possible this is an exact duplicate of another question, in which case my question should be closed.