I am required to find P$_{ij}^{(n)}$ for the P matrix of a 4 state Markov chain with states 0,1,2,3 as given below
$\begin{matrix} 1-p&&p&&0&&0\\0&&1-p&&p&&0\\0&&0&&1-p&&p\\0&&0&&0&&1\end{matrix}$
There were two options. One was to solve it by finding eigenvalues and then left and right eigenvectors...etc. Second choice was to use method of generating function
P(s) = I+sP+s$^2$P$^2$+s$^3$P$^3$...........
I chose to use the second method and got the (I-sP)$^{-1}$ as under:
\begin{matrix}\frac{1}{ps-s+1}&&\frac{-ps}{(ps-s+1)^2}&&\frac{p^2s^2}{(ps-s+1)^3}&&\frac{p^3s^3}{(s-1)(ps-s+1)^3}\\0&&\frac{1}{ps-s+1}&&\frac{-ps}{(ps-s+1)^2}&&\frac{-p^2s^2}{(s-1)(ps-s+1)^2}\\0&&0&&\frac{1}{ps-s+1}&&\frac{ps}{(s-1)(ps-s+1)}\\0&&0&&0&&\frac{-1}{(s-1)}\end{matrix}
Now to gather the coefficients of $s^n$ in each of the terms, I could do it for all terms of the inverse matrix except terms $a_{14}$, $a_{24}$ and $a_{34}$.
(a) Could I please get guidance to do this?
(b) Is there any other easier method to calculate the P$_{ij}^{(n)}$?