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?