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Math Help - Closed Formula for Generating Function

  1. #1
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    Closed Formula for Generating Function

    Hi Guys,

    Question: what is closed formula for generating function 1 /(1-2x^2)?

    Thanks
    Chicago Boy
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  2. #2
    MHF Contributor chisigma's Avatar
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    The Taylor expansion of f(x) around x=0 is...

    f(x) = \frac{1}{1- 2\cdot x^{2}} = 1 + 2\cdot x^{2} + 4\cdot x^{4} + \dots + (2\cdot x^{2})^{n} + \dots = \sum_{n=0}^{\infty} (2\cdot x^{2})^{n} (1)

    ... where the series converges for |x|< \frac{\sqrt{2}}{2}, so that f(x) = \frac{1}{1-2\cdot x^{2}} is the generating function of the sequence...

    a_{n}= 2^{\frac{n}{2}} n even, a_{n}=0, n odd

    Kind regards

    \chi \sigma
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