Determinant of a particular form of block matrix with commutating property

Question

What is the determinant of the block matrix $$\begin{pmatrix}A&J&\ldots&\ldots&J\\J&A&\ldots\ldots&J\\\vdots&\vdots&\ddots(2^m-times)&&\vdots\\\vdots&\vdots&&\ddots&\vdots\\J&J\ldots&&\ldots&A \end{pmatrix}$$ where $J$ is the all-ones matrix, and $A$ is some symmetric square matrix which has constant row sum, and hence commutes with $A$. When $m=1$ or $2$, the calculations are easy. But, as $m$ increases, the calculations become more difficult to collect. Specifically, some sort of alternating product comes into play. Any known formulae in this regard? What about the general case, that is if the iterations of $A$ are not a power of $2$? Thanks beforehand.