  A sequence $(a_n)_{n \geq 0}$ is Stieltjes moment sequence if it has the form
$a_n = \int_0^\infty x^n d\mu(x)$ for $\mu$ is a nonnegative measure on
$[0,\infty)$. It is known that $(a_n)_{n \geq 0}$ is a Stieltjes moment
sequence if and only if the matrix $H =[a_{i+j}]_{i,j \geq 0}$ is totally
positive, i.e., all its minors are nonnegative. We define a sequence of
polynomials in $x_1,x_2,\ldots,x_n$ $(a_n(x_1,x_2,\ldots,x_n))_{n \geq 0}$ to
be a Stieltjes moment sequence of polynomials if the matrix $H =[a_{i+j}
(x_1,x_2,\ldots,x_n)]_{i,j \geq 0}$ is $(x_1,x_2,\ldots,x_n)$-totally positive,
i.e., all its minors are polynomials in $x_1,x_2,\ldots,x_n$ with nonnegative
coefficients. The main goal of this paper is to produce a large class of
Stieltjes moment sequences of polynomials by finding multivariable analogues of
Catalan-like numbers as defined by Aigner.
