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Math Help - Another exponential generating function

  1. #1
    Member oldguynewstudent's Avatar
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    Another exponential generating function

    I am stuck on the following problem, perhaps one of the fine experts can show me the error of my ways?

    Use an EGF to solve the recurrence relation a_{0}=2 and a_{n}=na_{n-1}-n! for n\geq1.

    \sum_{n\geq1}\frac{a_{n}x^{n}}{n!}=\sum_{n\geq1}\f  rac{na_{n-1}x^{n}}{n!}-\sum_{n\geq1}\frac{n!x^{n}}{n!}

    E(x)-2=xE(x)-\frac{1}{1-x}+1

    E(x)(1-x)=\frac{-1}{1-x}+3=-\frac{1+3-3x}{1-x}

    E(x)=-\frac{4-3x}{(1-x)^{2}}=\frac{A}{1-x}+\frac{B}{(1-x)^{2}}; A=-3 B=-1

    E(x)=\frac{-1}{(1-x)^{2}}-\frac{3}{1-x}

    I cannot come up with the solution, any hints, nudges?
    Anyone want to give me a push in the right direction or perhaps off a cliff?
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  2. #2
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    Is it \frac{-1}{(1-x)^2} that's the bothering you?

    If so, differentiate \frac{1}{1-x} = \sum_{i=0}^{\infty} x^i
    to find a series for \frac{1}{(1-x)^2} .

    (I haven't checked your intermediate steps.)
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  3. #3
    Member oldguynewstudent's Avatar
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    A should have been +3 instead of -3. Just an arithmetic mistake.
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