  In this paper we show that many well-known counting coefficients, including
the Catalan numbers, the Motzkin numbers, the central binomial coefficients,
the central Delannoy numbers are Hausdorff moment sequences in a unified
approach. In particular we answer a conjecture of Liang at al. which such
numbers have unique representing measures. The smallest interval including the
support of representing measure is explicitly found. Subsequences of
Catalan-like numbers are also considered. We provide a necessary and sufficient
condition for a pattern of subsequences that if sequences are the Stieltjes
Catalan-like numbers, then their subsequences are Stieltjes Catalan-like
numbers. Moreover, a representing measure of a linear combination of
consecutive Catalan-like numbers is studied.
