First, we introduce the two-Motzkin-like number as the weight of vertically constrained Motzkin-like path with no leading vertical steps from (0, 0) to (n, 0) consisting of up steps, down steps, horizontal steps, vertical steps in the down direction and vertical steps in the up direction. Secondly, we provide sufficient conditions under which the two-Motzkin-like numbers (resp. the q-analogue of the two-Motzkin-like numbers) are Stieltjes moment sequences (resp. are q-Stieltjes moment sequences) and therefore infinitely log-convex sequences. As applications, on the one hand, we show that many well-known counting coefficients, including the central trinomial \(\left( {\begin{array}{c}2n\\ 2n\end{array}}\right) _{2}\) and pentanomial \(\left( {\begin{array}{c}2n\\ 4n\end{array}}\right) _{4}\) numbers of even indices respectively are Stieltjes moment sequences and, therefore, infinitely log-convex sequences in a unified approach. On the other hand, we prove that the sequence of polynomials of square trinomials \(\sum _{k=0}^{2n}\left( {\begin{array}{c}n\\ k\end{array}}\right) ^{2} _{2}q^{k}\) are q-Stieltjes moment sequence of polynomials. Finally, we provide a criterion for linear transformations and convolutions preserving Stieltjes moment sequences in more generalized triangular array.

首先，我们引入类两个Motzkin数作为垂直约束的类Motzkin路径的权重，其中没有从（0，0）到（n，0）的前导垂直步长，包括向上，向下，水平，垂直在向下方向上步进，在垂直方向上竖直。其次，我们提供了足够的条件，在该条件下，两个莫兹金像数（分别是两个莫兹金像数的q-模拟）是Stieltjes矩序列（分别是q -Stieltjes矩序列），因此无限对数-凸序列。作为应用，一方面，我们显示了许多众所周知的计数系数，包括中央三项式\（\ left（{\ begin {array} {c} 2n \\ 2n \ end {array}} \ right）_ {2} \）和五项式（{left（{\ begin {array} {c} 2n \\ 4n \ end {array}} \ right）_ {4} \）偶数索引分别是Stieltjes矩序列，因此，无穷大对数凸序列的统一方法。另一方面，我们证明了平方三项式的多项式序列\（\ sum _ {k = 0} ^ {2n} \ left（{\ begin {array} {c} n \\ k \ end {array} } \ right）^ {2} _ {2} q ^ {k} \）是多项式的q -Stieltjes矩序列。最后，我们提供了线性变换和卷积的准则，以更广义的三角阵列形式保留Stieltjes矩序列。