  The Catalan number sequence is one of the most famous number sequences in
combinatorics and is well studied in the literature. In this paper we further
investigate its fundamental properties related to the moment problem and prove
for the first time that it is an infinitely divisible Stieltjes moment sequence
in the sense of S.-G. Tyan. Besides, any positive real power of the sequence is
still a Stieltjes determinate sequence. Some more cases including (a) the
central binomial coefficient sequence (related to the Catalan sequence), (b) a
double factorial number sequence and (c) the generalized Catalan (or
Fuss-Catalan) sequence are also investigated. Finally, we pose two conjectures
including the determinacy equivalence between powers of nonnegative random
variables and powers of their moment sequences, which is supported by some
existing results.
