Aliniaeifard–Li identity for Catalan and tangent numbers

Let $[n]:=\{1,2,\ldots,n\}$. A set composition of $[n]$ is a list of mutually disjoint nonempty subsets $\phi_1,\ldots,\phi_{\ell}$ whose union is $[n]$, written $\phi=\phi_1/\cdots/\phi_{\ell}\vDash[n]$; let $\ell(\phi)$ denote its number of blocks. It is odd if every block has odd size. Let $C_n=\frac{1}{n+1}{2n\choose n}$ be the $n$th Catalan number, and let $E_n$ be defined by

$$\tan(x)+\sec(x)=\sum_{n\geq1}E_n\frac{x^n}{n!}.$$

For every even positive integer $\ell$, define

$$\mu_{\ell}=(-1)^{\frac{\ell}{2}-1}C_{\frac{\ell}{2}-1}.$$

Aliniaeifard and Li's identity. For any odd positive integer $n$,

$$\sum_{\phi\vDash[n]\atop\text{$\phi$ odd}}2^{n-\ell(\phi)}\mu_{\ell(\phi)+1}=(-1)^{\frac{n-1}{2}}E_n.$$

This is a combinatorial identity linking Catalan numbers with tangent numbers, arising in the study of the internal coproduct formula for peak algebra. The source presents it as a previously found result rather than as an unresolved claim.