I found an identity
I can prove this in a combinatorial way: from $n+1$ balls, choose one ball as a boundary and pick up $j$ balls from the left and $k-j$ balls from the right, for every possible boundary balls. And then I can prove this is identical to choosing $k+1$ balls from $n+1$ balls.
On the other hand, the wiki says that you can prove this by using negative binomial series expansion. I have tried few attempts but wasn't very successful. Could anyone share a proof using binomial series expansion?