The 2-log-convexity of the Apery Numbers

We present an approach to proving the 2-log-convexity of sequences satisfying three-term recurrence relations. We show that the Apery numbers, the Cohen-Rhin numbers, the Motzkin numbers, the Fine numbers, the Franel numbers of order 3 and 4 and the large Schroder numbers are all 2-log-convex. Numerical evidence suggests that all these sequences are k-log-convex for any $k\geq 1$ possibly except for a constant number of terms at the beginning.