  The divisibility and congruence of usual and generalized central trinomial
coefficients have been extensively investigated. The present paper is devoted
to analytic properties of these numbers. We show that usual central trinomial
polynomials $T_n(x)$ have only real roots, and roots of $T_n(x)$ interlace
those of $T_{n+1}(x)$, as well as those of $T_{n+2}(x)$, which gives an
affirmative answer to a open question of Fisk. We establish necessary and
sufficient conditions such that the generalized central trinomial coefficients
$T_n(b,c)$ form a log-convex sequence or a Stieltjes moment sequence.
