In Generatingfunctionology (2nd ed available free here https://www.math.upenn.edu/~wilf/DownldGF.html ), nCk for integer n < 0 is defined as
Then he asserts that:
I see why the summand would vanish for nonnegative integer n < k, because n*(n-1)...(n-n)...(n-k+1) = 0, but why for negative integer n? The text hasn't been corrected in the third edition, so I guess it's not an error? Or is it because nCk = nC(n-k), so if n is negative and k nonnegative n - k is negative, and that makes nC(n-k) = 0? Then how is that consistent with the earlier claim that nCk is nonzero for negative n and nonnegative k?