1. one has 2. 3. 4. Let Dn be the number of derangements of [n]. Prove that for n ≥ 1 Dn=nDn−1 + (−1)”. The generating function of the sequence (an)n is f(x)= = Use it to find a formula for an X (1-x)2(13x) Use the recurrence relation from Problem 1 to find the exponential generating function for the sequence (Dn)n. Recall that Do = 1. The following table gives the values of pk (n), the number of integer partitions of n into exactly k parts. Complete the 8th row and use it to find p(8), the number of integer partitions of 8. k 1 3 4 5 6 7 8 n 1 1 0 0 0 0 0 0 2 1 1 0 0 0 0 0 3 1 1 1 0 0 0 0 0 4 1 2 1 0 0 0 0 5 1 2 2 1 1 0 6 1 3 3 2 1 1 0 0 7 1 3 4 3 2 1 1 0 8 5. Consider the sequence (hn)n defined recursively as follows 6. ho = 1 h₁ = 2 hn=3hn-1+4hn-2 n ≥ 2. Find the generating function of this sequence. Consider the sequence (An)n defined recursively as follows Ao = [An+1 ¦ 1 = n Ai Σ n ≥ 0. (n - i)!' i=0 Find the generating function of this sequence.

1. Determine the asymptotic complexity of the function defined by the recurrence relation. Justify your solution using expansion/substitution. You may not use the Master Theorem as justification of your answer. Simplify and express your answer as (nk) or (nk log₂ n) whenever possible. If the recurrence is exponential just give exponential lower bounds. a) T(n) = T(n-4) + cn¹, T(0) = c' b) T(n) = T(n-3) + T(n-4) + c log₂ n, T(0) = c'

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Show that the given solution of the ODE is true. (Show working for the recurrence relationship)

DO PART (A) Show steps clearly with answer circled. No cursive if possible.

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Example 10.7.2. Let Pn D" (x" log x). Prove the recurrence relation Pn = n· Pn-1+ n – 1 Hence show that 1 = n! log x +1+ 1 +...+ 2 1 - -