Alternating Eulerian–Catalan identity for odd n

Let $n$ be odd. Write $vDash[n]$ for compositions of $[n]$, let $phi$ range over odd compositions, let $mu_{ell(phi)+1}$ denote the relevant Catalan-number coefficient, and let $E(n,k)$ be the number of permutations of $n$ with $k$ descents.

Alternating Eulerian–Catalan identity.

$$\sum_{\phi\vDash[n] \atop \phi\text{ is odd}}2^{n-\ell(\phi)}\mu_{\ell(\phi)+1}=\sum_{0\leq k\leq n-1}(-1)^kE(n,k).$$

This identity is reported as an unproved relation between Catalan numbers and Eulerian numbers and is described as irrelevant to the main result of the section.