A short proof of Mathar's 2021 recurrence conjecture for the Lehmer-Comtet diagonal A045406

For OEIS sequence A045406, the column-2 diagonal of the Lehmer-Comtet triangle A008296, R. J. Mathar contributed in September 2021 the conjectured order-2 P-recursive recurrence $$a(n)+(2n-7)a(n-1)+(n-4{)}^{2}a(n-2)=0,n\ge 5.$$ We give a short proof. Detlefs’s harmonic-number closed form $a(n)=(-1{)}^n(2H_{n-3}-3)(n-3)!$ for $n\ge 3$ collapses the left-hand side, after factoring out $(-1{)}^n(n-5)!(n-4)$, to a polynomial identity in $n$ with coefficient $H_{n-4}$. The $H_{n-4}$-coefficient simplifies to $(n-3)-(2n-7)+(n-4)=0$ (using $H_{n-3}=H_{n-4}+1\mathrm{/}(n-3)$ and $H_{n-5}=H_{n-4}-1\mathrm{/}(n-4)$); the constant remainder is $0$ for the same reason. The supplementary archive contains a SymPy script verifying both pieces symbolically, the e.g.f.\ expansion against the harmonic closed form, and Mathar’s recurrence numerically for $n=5,\dots ,5000$.