Is there any combinatorial interpretation or bijective proof for this identity
$$2C\_n=4{2n \choose n}-{2n+2 \choose n+1}$$
where $C\_n$ is the sequence of Catalan numbers?



Is there a combinatorial interpretation or bijective proof for this Catalan number identity?
There is an obvious bijective proof of the identity
$$ 2 \binom{2n}{n} + 2 \binom{2n}{n + 1} = \binom{2n+2}{n+1}$$
and also a bijective proof of
$$ 2\binom{2n}{n} - 2 \binom{2n}{n+1} = 2C\_n,$$
see the paragraph "second proof" [in the wiki page](https://en.wikipedia.org/wiki/Catalan_number#Second_proof). By combining the two, you get a bijective proof of your identity, in the form
$$ 4 \binom{2n}{n} =\binom{2n+2}{n+1} + 2C\_n .$$


