Sum over k of r-kt Choose k by r over r-kt by s-(n-k)t Choose n-k by s over s-(n-k)t/Proof 2

Proof
From Sum over $k$ of $\dbinom {r - k t} k$ by $\dfrac r {r - k t}$ by $z^k$:


 * $\displaystyle \sum_k A_k \left({r, t}\right) z^k = x^r$

and:


 * $\displaystyle \sum_k A_k \left({s, t}\right) z^k = x^s$

Hence:

Taking the $z^n$ coefficient: