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 November 8th, 2013, 07:24 PM #1 Newbie   Joined: Nov 2013 Posts: 3 Thanks: 0 Coefficient of Generating Function Hello, practicing for my upcoming exam and there's a question that I'm not familiar with the method to do it. Given the generating function : P(x) = (2-5x)/(1-5x+6x^2), how do we find an expression of Pn in terms of past values of Pn? Help please!!!
 November 8th, 2013, 07:43 PM #2 Global Moderator     Joined: Nov 2006 From: UTC -5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms Re: Coefficient of Generating Function I don't know what method you might have been taught, but let me see what I can do. The generating function is rational and its denominator is of degree 2, so there is a 2-term recurrence relation. That is, you can write P_{n} = a*P_{n-1} + b*P_{n-2} for some constants a and b. So just find the first few terms of the sequence and solve the resulting system of equations.
 November 8th, 2013, 07:51 PM #3 Newbie   Joined: Nov 2013 Posts: 3 Thanks: 0 Re: Coefficient of Generating Function I'm having trouble with finding the first coefficients.
 November 8th, 2013, 08:12 PM #4 Global Moderator     Joined: Nov 2006 From: UTC -5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms Re: Coefficient of Generating Function The terms of the generating function?
 November 8th, 2013, 08:59 PM #5 Newbie   Joined: Nov 2013 Posts: 3 Thanks: 0 Re: Coefficient of Generating Function yes
 November 9th, 2013, 09:53 AM #6 Global Moderator     Joined: Nov 2006 From: UTC -5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms Re: Coefficient of Generating Function It's just long division. Code:  2 + 5x + ... _________ 1-5x+6x^2 ) 2-5x -2+10x-12x^2 ------------ 5x-12x^2 -5x+25x^2-30x^3 -------------- ...

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