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1. Determine the sequence generated by each of the following generating functions.

(a) 2x3/1 - 5x2

(b) (5 + 3x)3

(c) 1 -x /1+x

2. For each of the following generating functions, find the coefficients of x15

(a) f(x) = 2x4/(1 + x)6

(b) h(x) = 1 /1 - 3x + x/(1+2x)(1- 3x)

3. Find a closed form generating function (i.e. not a series) for each of the following given sequences.

(a) 0, 0, 0, 2, 0, 2, 0, 2, 0......

(b) 2, 6, 18, 54, 162.....

(c) 1, 1, 0, 1, 1, 0, 1, 1, 0....

4. Jon has a jar with 70 Tuna TreatsTM and 30 Me-WowsTM to give out to his cats, Fluffy, Muffy and Zazzles. How many ways can Jon give out the treats so that Fluffy gets no Tuna TreatsTM, Muffy gets 4 or more Tuna TreatsTM and 5 or more Me- WowsTM, and Zazzles get an odd number of Tuna TreatsTM? (Hint: do the two treats separately to start).

5. Use generating functions to count the number of six-digit (positive) integers whose digits sum to 42. For ex, 978468 is a six-digit integer whose digits sum to

9 + 7 + 8 + 4 + 6 + 8 = 42.

6. Determine the closed form generating function for the sequence a0, a1, a2....., where an is the number of partitions of the nonnegative integer 'n' into

(a) even summands.

(b) distinct odd summands.

(c) summands that do not exceed 10 and occur at most 3 times.

(d) summands where summand 1 occurs at most once, summand 2 occurs at most twice, and in general, summand k occurs at most k times.

• Category:- Math
• Reference No.:- M9667

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