Suppose you have some $k$-regular tree centred on some root node $i$. The adjacency matrix representing this tree is $A$. Now for some $0<\alpha<1$ define $\tilde{A} = \frac{\alpha}{k} A$ so that the matrix is row (and column) sub-stochastic.
Supposing $\tilde{A}$ represents the transition probabilities between the transient states of a Markov chain, I'm interested in the fundamental matrix:
$$F = (I-\tilde{A})^{-1} = \sum_{m=0}^\infty \tilde{A}^m = I + \tilde{A} + \tilde{A}^2 + \ldots = I + \frac{\alpha}{k} A + \frac{\alpha^2}{k^2} A^2 + \ldots$$
Specifically, I'm interested in $F_{ji}$ - that is, the terms along the $i$-th column related to the root node related to the number of expected visits to the root node from nodes of various distance $d$.
Clearly, the only unknown term here are the elements of $A^m$. Given the symmetry and structure of this matrix, I assumed there were some well-established references and/or explicit derivations of these terms but I'm surprised I couldn't find anything. I even computed a few terms and plugged it into OEIS without any success.
Would love any references to explicit computations or approximations for these terms.