Catalan-like数包含了许多经典组合数列，在组合数学中起到重要的统一作用。此外，Catalan-like数建立了经典组合数列、递归矩阵以及经典分析之间的密切联系。因此Catalan-like数具有重要的研究价值。本项目将研究一些与Catalan-like数相关的组合问题，主要包括：.1.组合数列的Hankel转换。我们将运用递归矩阵和经典分析相结合的方法，研究与Catalan-like数相关的组合数列和多项式序列的Hankel转换。.2.Catalan-like数的组合性质研究。我们将利用Catalan-like数的性质，重点研究Engel关于Bell数相关数列凹性的猜想。.3.Catalan-like数的判定。我们将判定Apéry数是否为Catalan-like数，并进一步研究陈永川和Sokal分别关于Apéry数无穷对数凸性和Stieltjes moment性质的猜想。

Catalan-like numbers unify many well-known combinatorial sequences, including Catalan numbers, Motzkin numbers, Bell numbers and so on. Catalan-like numbers are closely related to recursive matrix and orthogonal polynomials. The project will be concerned with an important class of combinatorial sequences from the viewpoint of Catalan-like numbers, which contains:.1. We will study the Hankel transforms of both combinatorial sequences and polynomial sequences related to Catalan-like numbers by using the properties of recursive matrices and analysis. .2.We will observe the combinatorial properties of the sequences which belong to Catalan-like numbers. We will prove the conjecture about Bell numbers by using the properties of Catalan-like numbers. .3. We will study how to decide whether Apéry numbers belongs to Catalan-like numbers or not. And then we will consider the conjecture about Apéry numbers.