Abstract Many well-known Catalan-like sequences turn out to be Stieltjes moment sequences (Liang et al. (2016)). However, a Stieltjes moment sequence is in general not determinate; Liang et al. suggested a further analysis about whether these moment sequences are determinate and how to obtain the associated measures. In this paper we find necessary conditions for a Catalan-like sequence to be a Hausdorff moment sequence. As a consequence, we will see 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. We can also identify the smallest interval including the support of the unique representing measure. Since Hausdorff moment sequences are determinate and a representing measure for above mentioned sequences are already known, we could almost complete the analysis raised by Liang et al. In addition, subsequences of Catalan-like number sequences are also considered; we will see a necessary and sufficient condition for subsequences of Stieltjes Catalan-like number sequences to be Stieltjes Catalan-like number sequences. We will also study a representing measure for a linear combination of consecutive terms in Catalan-like number sequences.

中文翻译：

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类加泰罗尼亚数列和豪斯多夫矩列

摘要 许多著名的 Catalan-like 序列被证明是 Stieltjes 矩序列 (Liang et al. (2016))。然而，斯蒂尔杰斯矩序列通常是不确定的；梁等人。建议进一步分析这些矩序列是否确定以及如何获得相关测度。在本文中，我们找到了类 Catalan 序列是 Hausdorff 矩序列的必要条件。因此，我们将看到许多著名的计数系数，包括 Catalan 数、Motzkin 数、中心二项式系数、中心 Delannoy 数，都是 Hausdorff 矩序列。我们还可以确定最小间隔，包括唯一表示度量的支持。由于 Hausdorff 矩序列是确定的，并且上述序列的代表测度是已知的，我们几乎可以完成梁等人提出的分析。此外，还考虑了类加泰罗尼亚数列的子序列；我们将看到类 Stieltjes Catalan 数列的子序列是类 Stieltjes Catalan 数列的充要条件。我们还将研究类似 Catalan 的数字序列中连续项的线性组合的表示度量。