ws721 - x and of x 2 in the power series for 1 x-1 1 (2 x...

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Rob Bayer Math 55 Worksheet July 21, 2009 Generating Functions 1. What sequence is represented by each of the following generating functions? (a) ( x 2 + 1) 3 (b) 1 (1 - 2 x 2 ) (c) x 9 - 1 x - 1 2. Find a generating function for each of the following sequences: (a) a n = 2 (b) a n = 2 n (c) a n = n - 1 3. Find the coefficient of x 12 in 1 (1+ x ) 8 4. Find a generating function for the number of ways to make n cents using pennies, nickels, dimes, and quarters, where the order of the coins doesn’t matter. 5. Show that ± - 1 2 n ² = ( 2 n n ) ( - 4) n 6. Find a generating function for the number of solutions in non-negative integers to x 1 + x 2 + x 3 = k where 3 x 1 , 2 x 2 10 , 5 x 3 and x 3 is even. 7. Find a generating function for the number of solutions in non-negative integers to x 1 + 2 x 2 + 3 x 3 = k (Hint: consider new variables y 2 = 2 x 2 ,y 3 = 3 x 3 ) 8. (With calculus) (a) Explain why the coefficient of x n in the power series for G ( x ) is G ( n ) (0) n ! where G ( n ) ( x ) is the nth derivative of G (b) Use (a) to find the coefficients of
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Unformatted text preview: x and of x 2 in the power series for 1 x-1 1 (2 x +1) 3 (c) Find a formula in terms of derivatives for the number of solutions to x 1 + x 2 + x 3 + x 4 = 20 with x 2 even and x 3 divisible by 5 9. The generating functions we have been studying are called ordinary generating functions. Another class is called Exponential Generating Functions and are dened as follows: The Exponenetial Generating Function for the sequence { a n } is X n =0 a n n ! x n Note the division by n !. Thus, the exponenetial generating function for 1 , 1 ,... is x n n ! = e x . Use this fact to nd a closed form for the exponenetial generating function for each of the following sequences: (a) a n = 3 n (b) a n = (-1) n (c) a n = n + 1 (d) a n = 1 n +1 (Hint: re-index)...
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