Conjectural formula for the next-to-leading coefficient of path distance matrices

Let $P_n$ be a path on $n>5$ vertices. Define its characteristic polynomial by

$$\mathrm{CharPoly}_{\mathfrak{D}_{P_n}}(x)=\sum_{i=0}^{2n-3}a_i x^i.$$

Coefficient formula conjecture. The coefficient of $x^{2n-5}$ is

$$a_{2n-5}=-\frac{1}{6}(n-1)(n-2)(2n^2+6n-15).$$

The formula is proposed from computational data and is intended to estimate the coefficient needed for a lower-bound analysis of the peak location. No proof or resolution is given in the source.