I'm looking at the sequence $f(n)=\mathbf{1}^T C^{n} \mathbf{1}$ where $C$ is a $d\times d$ nonnegative matrix and $\mathbf{1}$ is a column vector of 1's.
I have a generating function $G_C$ for this sequence, how can I use it to obtain upper bounds on $f(n)$ for $n<d$?
The reason I can compute the generating function is because generating function of $(A+B)^n$ decomposes, making Woodbury matrix identity applicable
$$G_{A+B}=(I-x G_A B)^{-1} G_A$$