# ass2 - a n 4 Let a n = x n x 3-x 4 1-2 x x 2-x 6 Find a...

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MATH 239 Assignment 2 This assignment is due at noon on Friday, May 29, 2009, in the drop boxes opposite the Tutorial Centre, MC 4067. 1. Determine the following coefficient as a summation, where and m are non-negative integers: [ x n ](1 + 2 x ) - (1 - x 3 ) - m 2. Find the generating function for the number of compositions of n into an even number of parts, where each part is odd. 3. A certain brand of beer is sold in the form of either single cans, six packs, or boxes of 12 cans each. The store currently has 45 single cans, 20 six packs, and 100 boxes in stock. Let a n denote the number of ways to purchase n cans of beer. Find the generating function for
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Unformatted text preview: a n . 4. Let a n = [ x n ] x 3-x 4 1-2 x + x 2-x 6 . Find a linear recurrence equation for { a n } , together with enough initial conditions to uniquely specify { a n } . 5. For n ≥ 0, let b n = X ( a 1 ,a 2 ,...,a k ) a 1 a 2 ··· a k where the sum is over all compositions ( a 1 , a 2 , . . . , a k ) of n , and over all values of k . (a) Compute b n directly for n = 0 , 1 , 2 , 3 , 4. (b) Prove that b n = n + n X j =1 j b n-j . (c) Using part (b), or otherwise, prove that { b n } satisﬁes the linear recurrence equation b n = 3 b n-1-b n-2 . 1...
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