Catalan-like numbers and Stieltjes moment sequences

We provide sufficient conditions under which the Catalan-like numbers are Stieltjes moment sequences. As applications, we show that many well-known counting coefficients, including the Bell numbers, the Catalan numbers, the central binomial coefficients, the central Delannoy numbers, the factorial numbers, the large and little Schr\"oder numbers, are Stieltjes moment sequences in a unified approach.