# hw6 - n th term of at least one of them.) (5) (a) Show that...

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HOMEWORK #6, DUE ON 11/22 (1) Prove that the exponential generating function on the Bell numbers B n is exp(exp( x ) - 1). (2) Let c ( n, k ) = ( - 1) n - k s ( n, k ). Let a 0 , a 1 , . . . and b 0 , b 1 , . . . two arbitrary sequences. Prove that the following statements are equivalent. (a) n : b n = n k =0 S ( n, k ) a k (b) n : a n = n k =0 c ( n, k ) b k (3) (Exercise 14, Chapter 8) Let p be a Fxed positive integer. ±ind and verify a general formula for n s k =0 k p involving Stirling numbers of the second kind. (This means that from your formula it should be fast to compute the sum above, even for enourmous n , provided that you know all the Stirling numbers of the second kind. Even more precisely, the computational complexity of the evaluation of the sum may increase at most logarithmically as n grows.) (4) Let h n and g n be sequences that satisfy the following properties. i) h 0 = 1 ii) Δ n h 1 = g n iii) Δ 2 n g 0 = h n iv) Δ 2 n +1 g 0 = 0 Show that h n and g n are uniquely determined by these properties, and Fnd them. (This means that you should give a simple formula for the
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Unformatted text preview: n th term of at least one of them.) (5) (a) Show that the generating function of { p k ( n ) } ∞ n =0 is x k (1-x )(1-x 2 ) · · · (1-x k ) where p k ( n ) is the number of partitions of n into k terms. (b) Prove p odd ( n ) = p dist ( n ) using generating functions, where p odd ( n ) and p dist ( n ) are the number of partitions of n into odd terms, and distinct terms, respectively. 1 1 This was done in lecture in a diferent way without using generating Functions. When I was a students I hated those exercises when they told me how to do things, and another prooF wouldn’t be accepted. IF you Feel the same way, skip 5b), and do the Following exercise instead: ±ix t ≥ 0. Show that the sequence { p n-t ( n ) } ∞ n =0 will eventually become a constant. What is this constant f ( t )? 1...
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## This note was uploaded on 01/12/2012 for the course MATH 681 taught by Professor Wildstrom during the Fall '09 term at University of Louisville.

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