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Math Help - Exponential Generating Function

  1. #1
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    Exponential Generating Function

    I'm looking for the generating function of:
    an = (n^3)!
    which I know in other words is the closed formula for the summation attached.
    Any suggestions on how to do this one? Thanks!
    Attached Thumbnails Attached Thumbnails Exponential Generating Function-generating-function.jpg  
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  2. #2
    Senior Member
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    The n=0 term is 0, so you can work with the sum for n>0.

    For n>0, one n in the n^3 cancels with the n!.

    Pull an x out of the infinite sum. Now you've essentially got the same sum, but
    with the index variable as x-1. Rename this to y.

    The top now has a (y+1)^2. Expand this to y^2+2y+1, split the infinite sum into 3 infinite sums, and use this trick again.
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