How to do binomial coefficient

**othnin** · April 15th 2009, 08:06 PM

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

**chisigma** · April 16th 2009, 01:32 AM

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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