Problem: Given an integer $m$. We have \begin{align*} m &=m\\ m^2 & = m(m-1) + k\\ m^3 & = m(m-1)(m-2) + 3m(m-1) + k \end{align*} Now the matrix whose components (in column) are coefficients of convex combination that generates $m$, $m^2$ and $m^3$ is $$A_3 = \begin{bmatrix} 1 & 1 & 1\\ 0 & 1 & 3\\ 0 & 0 & 1 \end{bmatrix}. $$ Find the matrix $A_n$ and obserse what happens if $n$ goes to infinity (find infinity matrix?).
My attempt: For the question finding $A_n$, I have completed by induction and here is my solution \begin{align*} a_{ij} = \begin{cases} 0 & \text{ if } i>j\\ 1 & \text{ if } i = j\\ i\cdot a_{i,j-1} + a_{i-1,j-1} & \text{ if } i<j \end{cases} \end{align*}
For the moment, I have stucked here. I hope that you could give me some idea to deal with this problem.