Let A a superior triangular matrix with $0$ on its diagonal, and whose elements above the diagonal are different from $0$, of order $n \times n$

Question

I want to find an expression for the k power of this matrix:

\begin{equation} A= \begin{pmatrix} 0 & a_{12} & \cdots & \cdots& a_{1(n-1)} &a_{1n}\\ 0 & 0 & \cdots & \cdots &a_{2(n-1)} & a_{2n}\\ \vdots & \vdots & \vdots & \ddots & a_{3(n-1)} & a_{3n} \\ \vdots & \vdots & \vdots & \vdots & \ddots & \vdots\\ 0 & 0 & \cdots & 0 & 0 & 0 \end{pmatrix} \end{equation}

My first observation is that when $k\geq n \Rightarrow A^{k} = 0$

I have tried to generalize it, and for this I have taken n = 4 and k = 4, I have obtained an expression for the producer of certain terms, but it is difficult for me to find a general form for any k $\in \mathbb{N}$.

Any idea how to solve this exercise?