The coefficient formula for Leibniz graphs in the factorizing operator

Let $c_n$ be the coefficients in the factorizing operator

$$\Diamond=2\cdot\sum_{n\geqslant1}\frac{\hbar^n}{n!}\cdot c_n\cdot\mathcal{F}_{n-1}\bigl([\![{\mathcal{P},\mathcal{P}}]\!],\mathcal{P},\ldots,\mathcal{P}\bigr),$$

where $w_\Gamma$ is the weight of a Leibniz graph $\Gamma$ and $\#\!\operatorname{Aut}(\Gamma)$ denotes the order of its automorphism group. The coefficient formula for Leibniz graphs. For all $n\geqslant2$,

$$c_n=\frac{n}{3!}=\frac{n}{6},$$

and consequently the coefficient of the marker $\Gamma$ for the equivalence class $[\Gamma]$ is

$$2^n\cdot\frac{w_\Gamma}{\#\!\operatorname{Aut}(\Gamma)}.$$

This formula is supported by the experiments described in the source, but it remains to be explained how it follows from the $L_\infty$ condition; establishing that derivation would clarify the relation between the formality morphism and associativity of the star product.