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# PS04 - MATH 681 Problem Set#4 This problem set is due at...

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MATH 681 Problem Set #4 This problem set is due at the beginning of class on October 22 . 1. (10 points) For this problem, it will be helpful to note the following two power series expansions: e x + e - x 2 = 1 + x 2 2! + x 4 4! + x 6 6! + · · · e x - e - x 2 = x + x 3 3! + x 5 5! + x 7 7! + · · · (a) (5 points) Find an exponential generating function for the sequence { a n } of the number of ways to build an n -letter string consisting of As, Bs, Cs, and Ds such that there is at least one A, an even number of Bs, an odd number of Cs, and any number of Ds. (b) (5 points) Using your exponential generating function, find a formula for a n . 2. (20 points) A string is called “excellent” if it consists of any number of As, any number of Bs, and exactly one C. Let a n represent the number of excellent strings of length n . (a) (5 points) Construct an exponential generating function g ( x ) = n =0 a n x n n ! . (b) (5 points) Using casewise analysis on the first term of an excellent string, find a recurrence relation, with initial conditions, for a n .
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