If $|\rho|<1$, show that, when $n\to\infty$:
$$ \frac1{n-1} \sum_{k=1}^{n-1} \frac1{1-\rho^k-\rho^{n-k}} = 1 + \frac1n \frac{2\big(\psi_{\rho}(1)+\log\big(1-\rho)\big)}{\log \rho} + o\big(\tfrac1n\big) $$
where $\psi_a$ is the a-polygamma function. In particular $\psi_{\rho}(1) = \frac{\partial \log \Gamma_{\rho}(x)}{\partial x}\Big|_{x=1}$ where $\Gamma_a$ is the a-Gamma function.
FWIW, the question is related to this one.
