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Math Help - generating function problem

  1. #1
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    generating function problem

    If the generating function is (1-x^200)(1-x^240)/((1-x^10)(1-x^20)(1-x^25)). Then expand all the terms through x^100. How can I make it in the form of a recurrence relation hn. The initial conditions are given.
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  2. #2
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    Quote Originally Posted by kkkkkk View Post
    If the generating function is (1-x^200)(1-x^240)/((1-x^10)(1-x^20)(1-x^25)). Then expand all the terms through x^100. How can I make it in the form of a recurrence relation hn. The initial conditions are given.
    The generating function is

    (1-x^{200})(1-x^{240})(1-x^{10})^{-1}(1-x^{20})^{-1}(1-x^{25})^{-1}
    =(1-x^{200})(1-x^{240})\sum_{i=0}^\infty x^{10i} \sum_{j=0}^\infty x^{20j} \sum_{k=0}^\infty x^{25k}

    The first two factors, =(1-x^{200})(1-x^{240}), don't make any contribution to x^n with n < 200, so all you have to do is expand enough of

    \sum_{i=0}^\infty x^{10i} \sum_{j=0}^\infty x^{20j} \sum_{k=0}^\infty x^{25k}

    to get the powers of x up to x^{100}.

    I would start by expanding
    \sum_{j=0}^5 x^{20j} \sum_{k=0}^4 x^{25k}
    up to x^{100}.

    I don't know enough about what you want in the way of a recurrence to help you with that part of the problem.
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