Question. Do these identities involving even-index Catalan numbers have a known combinatorial interpretation? They look as though they should. I haven’t seen one in the literature (nor have I even seen these identities), but the combinatorics literature is large and I am small.

$ $ \sum_{a+b=n}C_{2a}C_{2b}=4^nC_n$ $

$ $ \sum_{a+b=n}C_{2a}{4b\choose 2b}=4^n{2n\choose n}$ $

Notes. I came across these identities while thinking about an unanswered question of Mike Spivey from 2012. I can define a bijection for the first one – I imagine the second can be done in a similar way – but it isn’t obviously a very nice bijection, and it uses Garsia-Milne involution. Here is a description of it, in the language of species.

If $ D$ is the species of Dyck paths with matching up/down pairs labelled (or binary plane trees with internal nodes labelled, etc.) then $ D=1+XD^2$ , where $ 1$ is the species of the empty set (sometimes denoted $ E_0$ ) and $ X$ is the species of singletons ($ E_1$ ). Split $ D=D’+D”$ , where $ D’$ represents paths of odd semilength and $ D”$ even, so we have the mutually recursive isomorphisms $ $ \begin{align}\tag{1}D”&=1+2XD’D”\\tag{2}D’&=X(D’^2 + D”^2)\end{align}$ $ (using the $ =$ sign to denote isomorphism). The aim is to define an isomorphism between $ D”^2$ and $ 1 + 4X^2D”^4$ . We shall do this by defining an isomorphism $ $ D”^2+Y=1 + 4X^2D”^4+Y$ $ for the species $ Y=(2XD’D”)^2$ , and appealing to the Garsia-Milne principle. Here is the isomorphism, from right to left:

$ $ \begin{align} 1 + 4X^2D”^4+Y&=1 + 4X^2D”^2(D’^2+D”^2)\ &=1 + 4XD’D”^2\tag{using 2}\ &=1 + 4XD’D”(1+2XD’D”)\tag{using 1}\ &=1 + 4XD’D” + 8(XD’D”)^2\ &=D”^2 + 4(XD’D”)^2\tag{using 1}\ &=D”^2 + Y \end{align}$ $