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I have this generating function:

$$\frac{1}{2}\, \left( {\frac {1}{\sqrt {1-4\,z}}}-1 \right) \left( \,{ \frac {1-\sqrt {1-4\,z}}{2z}}-1 \right)$$

and I know that $\frac {1}{\sqrt {1-4\,z}}$ is the generating function for the sequence $\binom {2n} {n}$, and $\frac {1-\sqrt {1-4\,z}}{2z}$ is the generating function for the sequence $\frac {1}{n+1}\binom{2n} {n}$.

Now, I thought that I could substitute those in there, and where they multiply I'll use a summation like this:

$$\frac{1}{2}\left( 1-\frac{1}{n+1}\binom{2n} {n}-\binom{2n} {n} + \sum_{k=0}^n \frac{1}{k+1} \binom{2k}{k}\binom{2(n-k)}{n-k} \right)$$

Could this be right? It doesn't seem to work when I try in Maple. What else could I do?

I already know that the end sequence will be $\binom{2n-1}{n-2}$ if this can help...

  • 0
    Be careful with your terminology: everywhere that you’ve written *series*, you should have *sequence*.2011-12-02
  • 0
    yep..I fixed it!2011-12-02
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    Note that $1$ (the constant function) is not the generating function for $1$ (the constant sequence), but rather for $\delta_{n}$ (the sequence with $a_0=1$ and $a_{n>0}=0$).2011-12-02
  • 0
    uh.. that's right. So what should I put there? some function of n that gives 1 on n=0 and 0 the rest of the time? What function is that?2011-12-02
  • 0
    @Adam: I like the [Iverson bracket](http://en.wikipedia.org/wiki/Iverson_bracket) for that purpose.2011-12-05

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