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I am studying some problems in Free Probability and need to prove a relation between the Catalan numbers

[tex]C_n = \frac{1}{n+1}\left(\stackrel{2n}{n}\right)[/tex]

The relation reads:

[tex]1 = \sum_{l=1}^{k}{(-1)^{l-1}C_{l-1}\left(\stackrel{l+k-1}{k-l}\right)}\quad\quad\forall\quad 1\leq k\in\mathbb{N}[/tex]

Does anybode have an idea how to prove this? Numerically, I convinced myself that it is true for [tex]n\leq 100 000[/tex] or so

I tried induction but it didn't work out.

Thanks

Pere

Btw.: Is there a better way to display binomial coefficients....? I used \stackrel ...

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# Relation between Catalan numbers

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