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Generating functions and sums with binomial coefficients

  1. Mar 1, 2012 #1
    1. The problem statement, all variables and given/known data
    Show that the generating function [itex]A(x) = \sum_n a_n x^n[/itex] of

    [tex]a_n = \sum_{k=0}^n {n+k \choose 2k} 2^{n-k}[/tex]

    satisfies

    [tex]A(x) = \frac{1-2x}{4x^2-5x+1}[/tex]


    2. Relevant equations



    3. The attempt at a solution
    A hint was given to "interchange the sums". After doing that, I don't see how to proceed. I also obtained the coefficients by partial fractions on A(x) but it's definitely non-trivial to show these are a_n. Thanks for any help.
     
    Last edited: Mar 1, 2012
  2. jcsd
  3. Mar 1, 2012 #2

    jbunniii

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    Can you show what you got after interchanging the sums?
     
  4. Mar 1, 2012 #3
    This one is done. Thanks for taking a look!
     
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