Higher order log-concavity in Euler’s difference table

For 0≤k≤n, let enk be the entries in Euler’s difference table and let dnk=enk/k!. Dumont and Randrianarivony showed enk equals the number of permutations on [n] whose fixed points are contained in {1,2,…,k}. Rakotondrajao found a combinatorial interpretation of the number dnk in terms of k-fixed-points-permutations of [n]. We show that for any n≥1, the sequence {dnk}0≤k≤n is essentially 2-log-concave and reverse ultra log-concave. ► In this paper, we study the higher order log-concavity of {dnk}0≤k≤n, where dnk=enk/n! and enk are the entries in Euler’s difference table. ► We show that the sequence {dnk}0≤k≤n is essentially 2-log-concave for any n≥1. ► We also show that the sequence {dnk}0≤k≤n is reverse ultra log-concave for any n≥1.