A Sequence of Binomial Coefficients Related to Lucas and Fibonacci Numbers

A positive real sequence (ak) n k=0 is said to be unimodal if there exist integers k0, k1, 0 ≤ k0 ≤ k1 ≤ n such that a0 ≤ a1 ≤ · · · ≤ ak0 = ak0+1 = · · · = ak1 ≥ ak1+1 ≥ · · · ≥ an. The integers l, k0 ≤ l ≤ k1 are called the modes of the sequence. If k0 < k1 then (ak)k=0 is said to have a plateau of k1 − k0 + 1 elements; if k0 = k1 then it is said to have a peak. A real sequence is said to be logarithmically concave (log-concave for short) if ak ≥ ak−1ak+1, 1 ≤ k ≤ n− 1 (1) If the inequalities in (1) are strict, then (ak) n k=0 is said to be strictly log-concave (SLC for short). A sequence is said to be have no internal zeros if i < j , ai 6= 0 and aj 6= 0, then ak 6= 0 for i ≤ k ≤ j. A log-concave sequence with no internal zeros is obviously unimodal, and if it is SLC, then it has at most two consecutive modes. The following result is sometimes useful in proving log-concavity. For a proof of this theorem, see Hardy and Littlewood [5].