I am trying to prove this equation directly, but so far without success (unless $B=1$).
$$\sum_{k=0}^{A} C_{A+B}^{B+k}C_{B+k-1}^{k}(-1)^{k}=1$$
I found it can be transformed into beta function, as
$$\int_0^1x^{B-1}(1-x)^{A}dx=\frac{A!(B-1)!}{(A+B-1)!}$$
but is there any other more straight forward way to prove it?