Unformatted text preview: x n ](1-2 x )-2 and, using this, compute the average. 4. Let A ( x ) and B ( x ) be series with constant terms equal to 1 such that B ( x ) = A ( x ) 3 . Note that if C ( x ) is the series ∑ n ≥ c n x n , then its derivative C ( X ) is ∑ n ≥ ( n + 1) c n + 1 x n . (a) (2 points) Using the usual rules from Calculus, show that A ( x ) B ( x ) = 3 A ( x ) B ( x ). (b) (2 points) Assuming that [ x k ] A ( x ) = a k and [ x k ] B ( x ) = b k , compute formulas for the coefﬁcient of x n in A ( x ) B ( x ) and in A ( x ) B ( x ). (c) (5 points) Apply the previous material with B ( x ) = (1-x )-1 to compute the ﬁrst three coefﬁcients of (1-x )-1/3 . 5. (5 points) Solve the recurrence equation: b n-3 b n-1-4 b n-2 + 12 b n-3 = where b = 1, b 1 = 7, b 2 = 9....
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- Spring '09
- Math, Recurrence relation, Generating function, binary strings, ﬁrst ﬁve coefﬁcients