Summary: Let $L(n,k) = n / (n-k) C(n-k, k)$. We prove that all the zeros of the polynomial $L_n(x)= sum L(n,k)$x^k are real. The sequence $L(n,k)$ is thus strictly log-concave, and hence unimodal with at most two consecutive maxima. We determine those integers where the maximum is reached. In the last section we prove that $L(n,k)$ satisfies a central limit theorem as well as a local limit theorem.

11B39, 11B65

Fibonacci number, log-concave sequence, limit theorems, Lucas number, polynomial with real zeros, unimodal sequence