Unformatted text preview: a n . 4. Let a n = [ x n ] x 3x 4 12 x + x 2x 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 nj . (c) Using part (b), or otherwise, prove that { b n } satisﬁes the linear recurrence equation b n = 3 b n1b n2 . 1...
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 Spring '09
 M.PEI
 Combinatorics, Integers, Recurrence relation, Generating function, linear recurrence equation, recurrence equation bn

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