Just wondering if anyone knows any references in the literature to bijections corresponding to the following simple generating function identities. Let $B(z)=\dfrac{1}{\sqrt{1-4z}}$ and $C(z)=\dfrac{1-\sqrt{1-4z}}{2z}$, the generating functions of the central binomial coefficients and Catalan numbers, respectively. I'm looking for bijections corresponding to the following identities: $$B(z)B(-z)=B(4z^2),$$ $$\frac{C(z)+C(-z)}{2}=B(z)(1-2zC(4z^2))=B(-z)(1+2zC(4z^2))$$ and, taking the product of the last two expressions in the second identity and using the first identity and the fact that $1-zC^2=C/B$, $$\left(\frac{C(z)+C(-z)}{2}\right)^2=C(4z^2).$$ Thanks.


*Update:* In a similar vein, one can try to prove other such identities. For example, let $E(z)=\dfrac{1}{\sqrt{1-2z-3z^2}}$ be the generating function of the central trinomial coefficients, then $$E(4z)=B(-z)B(3z) \quad \text{and} \quad E(2z)E(-2z)=B(z^2)B(9z^2).$$



Proofs of some combinatorial identities
Partial answer: Your first identity is
\begin{equation}
\sum\limits\_{k=0}^n \left(-1\right)^k \dbinom{2k}{k} \dbinom{2\left(n-k\right)}{n-k} = \left[n \text{ is even}\right] 2^n \dbinom{n}{n/2} ,
\end{equation}
where I am using the Iverson bracket notation. (That is, $[\mathcal{A}]$ denotes the truth value of a statement $\mathcal{A}$.) 


This identity is proven in: [Michael Z. Spivey, *A Combinatorial Proof for the Alternating Convolution of the Central Binomial Coefficients*](http://mathcs.pugetsound.edu/~mspivey/AltConvCenBinFinal.pdf). This paper actually [arose](https://math.meta.stackexchange.com/a/14997/) from [an m.se question](https://math.stackexchange.com/questions/80092).


Disclaimer: I have not read the proof.


