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This is the Exam of Discrete Mathematics which includes Recurrence Relation, Space Is Available, Answer, Number of Ways, Sum of Odd Integers, By Hand, Solution, Various Walks, Provided etc. Key important points are: Counted, Convolution, Sequence, Generating Functions F, Coefficient, Recurrence Relation, Sequence, Generating Function, Nonnegative Integers, Number of Solutions
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MACM 201 Summer 2007 Instructor: Robert ˇS´amal July 4, 2007, 12:30 – 13:
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4 2
[2] (b) Explain why p(4) = 5. (That is, list all objects that are counted by p(4).)
[4] (c) Let an = 2 and bn = n for all n ≥ 0. Suppose that the sequence c 0 , c 1 , c 2 ,... is the convolution of a 0 , a 1 , a 2 ,... and b 0 , b 1 , b 2 ,.... What is cn?
an − 2 an− 1 = 3n^ (n ≥ 1), a 0 = 3.
Find the generating function for the sequence a 0 , a 1 , a 2 ,....
a + b + c = 100 ,
where a, b, c are nonnegative integers such that
[3] (a) Express the answer as a coefficient of some power of x in an appropriate generating function f (x).
[4] (b) Express f (x) as a polynomial (1−x (^2) ) 3.
[3] (c) Use the binomial theorem to get the final answer.