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Alldiff Congruences, Graph Theoretic Method, and Beyond
Published in Khodakhast Bibak, Restricted Congruences in Computing, 2020
If (n, k) = 1, then (6.3.1) is independent of b and simplifies as Nn(k)=k!n(nk). (Of course, this can also be proved directly.) If in addition we have n = 2k + 1, then Pn(k)=1k!Nn(k)=12k+1(2k+1k)=1k+1(2kk), which is the Catalan number.
Discrete Mathematics
Published in Dan Zwillinger, CRC Standard Mathematical Tables and Formulas, 2018
The Catalan numbers are Cn=1n+12nn≈4nn3/2 $ C_{n} = \frac{1}{n + 1}\left( {\begin{array}{*{20}c} {2n} \\ n \\ \end{array} } \right) \approx \frac{{4^{n} }}{{n^{3/2} }} $ for large n. One recurrence
Combinatorics
Published in Paul L. Goethals, Natalie M. Scala, Daniel T. Bennett, Mathematics in Cyber Research, 2022
Remarkably, Catalan trees and binary trees are both enumerated by the same numbers; the numbers 1n+1(2nn) are called the Catalan numbers. At a high level, the two recurrences above can be described as: an object is either empty or is an atom with two smaller objects, or, an object is an atom with some (possibly zero) ordered smaller objects, respectively. These two ideas are ubiquitous within combinatorial structures, and where we find them, we find the Catalan numbers.
The inverse versine function and sums containing reciprocal central binomial coefficients and reciprocal Catalan numbers
Published in International Journal of Mathematical Education in Science and Technology, 2022
Recall the nth Catalan number is defined by the recursion for with . An explicit expression for the nth Catalan number is known, it being given by Stanley (2015, p. 4, Eq. 1.6) From (25) series containing the reciprocal of the Catalan numbers can thus be readily found from series containing the reciprocal of the central binomial coefficients given in Section 3.1.