  The Catalan-Larcombe-French sequence $\{P_n\}_{n\geq 0}$ arises in a series
expansion of the complete elliptic integral of the first kind. It has been
proved that the sequence is log-balanced. In the paper, by exploring a
criterion due to Chen and Xia for testing 2-log-convexity of a sequence
satisfying three-term recurrence relation, we prove that the new sequence
$\{P^2_n-P_{n-1}P_{n+1}\}_{n\geq 1}$ are strictly log-convex and hence the
Catalan-Larcombe-French sequence is strictly 2-log-convex.
