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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) = (25x)/(15x+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 2term recurrence relation. That is, you can write P_{n} = a*P_{n1} + b*P_{n2} 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 + ... _________ 15x+6x^2 ) 25x 2+10x12x^2  5x12x^2 5x+25x^230x^3  ... 

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