TITLE: Alternating sign Vandermonde convolution
QUESTION [11 upvotes]: The well-known Vandermonde convolution gives us the closed form $$\sum_{k=0}^n {r\choose k}{s\choose n-k} = {r+s \choose n}$$
For the case $r=s$, it is also known that $$\sum_{k=0}^n (-1)^k {r \choose k} {r \choose n-k} = (-1)^{n/2} {r \choose n/2} \quad [n \mathrm{\ is\ even}]$$
When $r\not= s$, is  there a known closed form for the alternating Vandermonde sum? $$\sum_{k=0}^n (-1)^k {r \choose k} {s \choose n-k}$$

REPLY [4 votes]: You can use the Zeilberger algorithm to find a recurrence for the sum and then use Petkovsek's algorithm to verify that there are no hypergeometric (i.e. essentially products of factorials) closed formulas for the general case.
So, you cannot expect to do better than the expressions proposed in the other answers.