Abstract The Motzkin numbers count the number of lattice paths which go from ( 0 , 0 ) to ( n , 0 ) using steps ( 1 , 1 ) , ( 1 , 0 ) and ( 1 , − 1 ) and never go below the x -axis. Let M n , k be the number of such paths with exactly k horizontal steps. We investigate the analytic properties of various combinatorial triangles related to the Motzkin triangle [ M n , k ] n , k ≥ 0 , including their total positivity, the real-rootedness and interlacing property of the generating functions of their rows, and the asymptotic normality (by central and local limit theorems) of these triangles. We also prove several identities related to these triangles.

中文翻译：

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与 Motzkin 数相关的组合三角形的解析性质

摘要 Motzkin 数计算从 ( 0 , 0 ) 到 ( n , 0 ) 使用步骤 ( 1 , 1 ) , ( 1 , 0 ) 和 ( 1 , − 1 ) 并且永远不会低于 x 的点阵路径的数量。 -轴。令 M n , k 为具有恰好 k 个水平步长的此类路径的数量。我们研究了与 Motzkin 三角形 [ M n , k ] n , k ≥ 0 相关的各种组合三角形的解析性质，包括它们的总正性、它们行的生成函数的实根性和交错性质，以及渐近正态性（通过中心和局部极限定理）这些三角形。我们还证明了与这些三角形相关的几个恒等式。