Exercise III: Catalan numbersFor n e N, denote by c, to be the number of ways to form a "mountain range" with nupstrokes (U) and n downstrokes (D) that all stay above a horizontal line. For instance:for n = 1, only UD is allowed, so c = 1;• for n = 2, only UUDD and UDUD are allowed, so c2 = 2.1. Check that c3 = 5 by writing down or drawing all possible options.2. Consider the power series+00g(x) = 2n=0(by definition co =1) called the generating function of the sequence (en). Justifythat cn < 4" for each n, and deduce that the radius of convergence of g is at least1/4.3. It can be show that for r E (-1/4, 1/4),g(x) = 1+ xg(r)?and therefore1- VI4.xg(x) =2.xUse this formula and the known power series of V1+ x to write the first terms ofthe power series expansion of g, in the formg(x) = co + c1x + c2x2 + C3x + c4x* + ·..Show how you obtain a few terms, but you do not need to show all computationsand you can use a calculator for fractions. Check that you recover co, C1, C2, C3, anddeduce the value of c4.4. (Bonus, ungraded, for fun) Check your result by writing down or drawing all possiblepaths for n = 4.

Answer: Our guidelines is answer the first three question from the first question. I have given…

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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