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Math Help - How to do binomial coefficient

  1. #1
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    How to do binomial coefficient

    Okay,
    One more question. This section of the book is pretty spotty.
    Find a closed for for the generating function for the sequence a(n) = \binom{n}{2} , for n=0, 1, 2.

    There is the table for generating functions and I was looking at 1/(1+x)^n but I don't really think that is correct.
    Also I tried to use the math tags to make it more legible for once. Hopefully I least got that right.

    Thanks
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  2. #2
    MHF Contributor chisigma's Avatar
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    By definition the generating function of a sequence a_{n} is...

    g(x)= \sum_{n=0}^{\infty} a_{n}\cdot x^{n}

    In your case is...

    a_{n}= \binom{n}{2}= \frac{n\cdot(n-1)}{2}

    ... so that...

     g(x)= \sum_{n=2}^{\infty} \frac{n\cdot(n-1)}{2} \cdot x^{n} = \frac{x^{2}}{2} \frac{d^{2}}{dx^{2}} \sum_{n=0}^{\infty} x^{n}= \frac{x^{2}}{2} \frac{d^{2}}{dx^{2}} (\frac {1}{1-x}) = \frac{x^{2}} {(1-x)^{3}}

    Kind regards

    \chi \sigma
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