I am trying to compute the first column $P_{k,1}^n$ of the power of matrix $P^n$ where $P$ is a lower bidiagonal matrix with terms : $$P_{i,j} = \left\{\begin{array}{cc} i\alpha & \text{if }i=j,\\ (1-(i-1)\alpha) & \text{if }(i-1)=k,\\ 0 & \text{otherwise}. \end{array}\right.$$

I managed to get the problem down to : $P_{k,1}^n=C_{n,k}\alpha^{n-k+1}\prod_{i=1}^{k-1}(1-k\alpha)$ where $C_{1,1}=C_{1,2}=1$ and $C_{n,0}=0$ and $C_{n,k}=kC_{n-1,k}+C_{n-1,k-1}$. It is extremely close to the Pascal rule and I was wondering if there was a way to get a closed form formula for $C_{n,k}$.