The Catalan numbers
نویسندگان
چکیده
E. Catalan stated in 1874 that the numbers (2m)! (2n)!/m! n! (m+n)! are integers. When m = 0 these numbers are the middle binomial coefficients ( 2n n ) . When m = 1 they are twice the Catalan numbers. In this paper, we give a combinatorial interpretation for these numbers when m = 2.
منابع مشابه
Some Properties of the Fuss–catalan Numbers
In the paper, the authors express the Fuss–Catalan numbers as several forms in terms of the Catalan–Qi function, find some analytic properties, including the monotonicity, logarithmic convexity, complete monotonicity, and minimality, of the Fuss–Catalan numbers, and derive a double inequality for bounding the Fuss–Catalan numbers.
متن کاملThe Super Patalan Numbers
We introduce the super Patalan numbers, a generalization of the super Catalan numbers in the sense of Gessel, and prove a number of properties analogous to those of the super Catalan numbers. The super Patalan numbers generalize the super Catalan numbers similarly to how the Patalan numbers generalize the Catalan numbers.
متن کاملParametric Catalan Numbers and Catalan Triangles
Here presented a generalization of Catalan numbers and Catalan triangles associated with two parameters based on the sequence characterization of Bell-type Riordan arrays. Among the generalized Catalan numbers, a class of large generalized Catalan numbers and a class of small generalized Catalan numbers are defined, which can be considered as an extension of large Schröder numbers and small Sch...
متن کاملA 55 Integers 10 ( 2010 ) , 771 - 792 a Left Weighted Catalan Extension
There are many representations of the Catalan numbers. In this article, we will examine the ballot problem and extend it beyond the standard p-Catalan extension to a left weighted extension. This extension reduces naturally to the p-Catalan numbers (including the standard 2-Catalan numbers). We will then give a variety of examples of the extension for other representations of the Catalan number...
متن کاملStaircase tilings and k-Catalan structures
Many interesting combinatorial objects are enumerated by the k-Catalan numbers, one possible generalization of the Catalan numbers. We will present a new combinatorial object that is enumerated by the k-Catalan numbers, staircase tilings. We give a bijection between staircase tilings and k-good paths, and between k-good paths and k-ary trees. In addition, we enumerate k-ary paths according to D...
متن کاملSome properties of the Catalan–Qi function related to the Catalan numbers
In the paper, the authors find some properties of the Catalan numbers, the Catalan function, and the Catalan-Qi function which is a generalization of the Catalan numbers. Concretely speaking, the authors present a new expression, asymptotic expansions, integral representations, logarithmic convexity, complete monotonicity, minimality, logarithmically complete monotonicity, a generating function...
متن کامل