Exercise III: Catalan numbers For n e N, denote by c, to be the number of ways to form a "mountain range" with n upstrokes (U) and n downstrokes (D) that all stay above a horizontal line. For instance: • for n = 1, only UD is allowed, so c1 = 1; • for n = 2, only UUDD and U DUD are allowed, so c2 = 2. 1. Check that c3 = 5 by writing down or drawing all possible options. 2. Consider the power series „a" 3 = (x)6 n=0 (by definition co = 1) called the generating function of the sequence (en). Justify that Cn < 4" for each n, and deduce that the radius of convergence of g is at least 1/4.
Exercise III: Catalan numbers For n e N, denote by c, to be the number of ways to form a "mountain range" with n upstrokes (U) and n downstrokes (D) that all stay above a horizontal line. For instance: • for n = 1, only UD is allowed, so c1 = 1; • for n = 2, only UUDD and U DUD are allowed, so c2 = 2. 1. Check that c3 = 5 by writing down or drawing all possible options. 2. Consider the power series „a" 3 = (x)6 n=0 (by definition co = 1) called the generating function of the sequence (en). Justify that Cn < 4" for each n, and deduce that the radius of convergence of g is at least 1/4.
Computer Networking: A Top-Down Approach (7th Edition)
7th Edition
ISBN:9780133594140
Author:James Kurose, Keith Ross
Publisher:James Kurose, Keith Ross
Chapter1: Computer Networks And The Internet
Section: Chapter Questions
Problem R1RQ: What is the difference between a host and an end system? List several different types of end...
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