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Conjectured exact formula for the gap between the largest and second-largest vertex degrees

Prove that for all n the difference between the largest and second-largest vertex degrees satisfies M^{(0)}(n) − \widetilde{M}^{(0)}(n) = \left. \dfrac{d^{\,n-2}}{dx^{\,n-2}}\left(\dfrac{e^x}{1-x}\right)\right|_{x=0} (equation (375)).

References

We guess that equation~#1{375} is valid for an arbitrary $n$, which leads to the asymptotics:

— Deterministic simplicial complexes (Dorogovtsev et al., 10 Jul 2025) in Section 2 (Unconstrained growth), Upper degrees; around Eq. (375)