Consider a sequence of matrices defined recursively as follows:
$P(0) = I_{n}$
$P(1) = M$
$P(t) = \beta \,P(t-1) M + (1-\beta)\, P(t-2)$
where $M$ is the transition matrix of a random walk on an $n$-cycle with loop probability of $1/3$ on each node, and $\beta = \frac{2}{1+\sqrt{1-\lambda^2}}$ with $\lambda$ being the second-largest eigenvalue of $M$ (This is a special case of the recursion given by the second-order Richardson method in numerical analysis).
$P(t)$ can have negative entries, and my aim is to obtain a formula for the sum of negative entries in an arbitrary row, as a function of $n$ and $t$.