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Unformatted text preview: 1. Find the generating functions for the following sequences: , 1 , 4 , 13 , . . . ( a n = 3 n 1 2 ) 7 , 10 , 15 , 22 , . . . ( a n = n 2 + 2 n + 7 ) 3 , 3 , 3 , 3 , 4 , 5 , 6 , 7 , . . . ( a n = n for n 3 , a = a 1 = a 2 = 3 ) 2 , 1 , 5 , 7 , 17 , . . . ( a n = 2 n + 1 for n even, a n = 2 n 1 for n odd) 1 1 3 x is the GF for a n = 3 n , 1 1 3 x 1 1 x is the GF for a n = 3 n 1, thus the result is 1 2 parenleftBig 1 1 3 x 1 1 x parenrightBig . The GF for a n = n 2 is x ( x +1) (1 x ) 3 , the GF for a n = n is x (1 x ) 2 , the result is x ( x +1) (1 x ) 3 + 2 x (1 x ) 2 7 1 x . The GF is obtained from the GF for a n = n by adding the GF for 3 , 2 , 1 , , , , . . . , i.e., x (1 x ) 2 + 3 + 2 x + x 2 . a n = 2 n + ( 1) n , thus the GF is 1 1 2 x + 1 1+ x . 2. Find the generating function for the number of partitions of n to distinct even summands....
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 Spring '09
 MarniMishna

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