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

TL;DR

This paper provides a concise proof of Mathar's 2021 recurrence conjecture for a specific OEIS sequence, using harmonic number closed forms and symbolic verification tools.

Contribution

It offers a short, rigorous proof of the recurrence conjecture, confirming the harmonic number closed form and validating results with symbolic computation.

Findings

Confirmed Mathar's recurrence conjecture for the sequence

Derived a harmonic number closed form for the sequence

Validated the proof with symbolic verification and numerical checks

Abstract

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 \[ a(n) + (2n-7)\,a(n-1) + (n-4)^{2}\,a(n-2) \;=\; 0,\qquad n \ge 5. \] We give a short proof. Detlefs's harmonic-number closed form $a(n) = (-1)^n (2 H_{n-3} - 3)(n-3)!$ 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 $H_{n-3} = H_{n-4} + 1/(n-3)$ and $H_{n-5} = H_{n-4} - 1/(n-4)$); 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 $n = 5, \ldots, 5000$.