Good afternoon. My question is, if there is any known formula for the expression $A^n \mathbf{x}^n + A^{n-1} \mathbf{x}^{n-1} + \ldots+ A \mathbf{x} + I,$ where $A$ is a matrix and $\mathbf{x}$ is a column vector. By $\mathbf{x}^n$, I denote the element-wise $n$-th power of the column vector $\mathbf{x}$.
What I am looking for is something like a closed-form formula. It seems to be likely that something like that exists, since the expression resembles the geometric sum.
Thanks a lot in advance.