  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.

Combinatorics;Number Theory