Assuming we have two dense matrices $A \in \mathbb{R}^{m\times m}, B \in \mathbb{R}^{m\times n}$, is there a smart way to compute all entries of the series $A^1 B, A^2 B, A^3 B, \dots, A^k B$ up to some k? In general the matrices are arbitrary, but I'd be willing to accept some not-super-restrictive structure if that would make the problem easier, e.g. diagonalizable or PSD.
The minimally-smart way would be to just do it incrementally, which leads to $O ( k \max(m,n)^\omega )$, where $2 \le \omega \le 2.373$ is the matrix multiplication constant. I'm curious if there's some structure we could exploit to make that cheaper. I'm interested in practical applications so even non-asymptotic improvements could be fine.
