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 $$E(4z)=B(-z)B(3z) \quad \text{and} \quad E(2z)E(-2z)=B(z^2)B(9z^2).$$

49(2012), 391-396) seems to be very closely related. Except maybe actually your question was inspired by that paper? $\endgroup$ – მამუკა ჯიბლაძე Oct 1 '17 at 8:57