1 evaluate the sequence generated by each of the

Question

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

2. For each of the subsequent generating functions, nd the coecient of x¹⁵ .

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, Fluy, Muy and Zazzles. How many ways can Jon give out the treats so that Fluy gets no Tuna TreatsTM, Muy gets 4 or more Tuna TreatsTM and 5 or more Me- WowsTM, and Zazzles get an odd number of Tuna TreatsTM?

5. Use generating functions to count the number of six-digit (positive) integers whose digits sum to 42. For example, 978468 is a six-digit integer whose digits sum to 9 + 7 + 8 + 4 + 6 + 8 = 42.

6. Evaluate 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.