Geodesic
A sequence of binomial coefficients related to Lucas and Fibonacci numbers
Journal of integer sequences, Tome 6 (2003) no. 2
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.
Classification : 11B39, 11B65
Keywords: Fibonacci number, log-concave sequence, limit theorems, Lucas number, polynomial with real zeros, unimodal sequence 
@article{JIS_2003__6_2_a0,
     author = {Benoumhani,  Moussa},
     title = {A sequence of binomial coefficients related to {Lucas} and {Fibonacci} numbers},
     journal = {Journal of integer sequences},
     year = {2003},
     volume = {6},
     number = {2},
     zbl = {1142.11311},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/JIS_2003__6_2_a0/}
}
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Benoumhani,  Moussa. A sequence of binomial coefficients related to Lucas and Fibonacci numbers. Journal of integer sequences, Tome 6 (2003) no. 2. http://geodesic.mathdoc.fr/item/JIS_2003__6_2_a0/