Higher order log-concavity in Euler’s difference table

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