A short proof of Mathar's 2020 recurrence conjecture for the generalized-Stirling sequence A001711

TL;DR

This paper provides a concise proof confirming Mathar's 2020 recurrence conjecture for the generalized-Stirling sequence A001711, utilizing harmonic number identities and symbolic verification.

Contribution

It offers a one-page proof of the recurrence conjecture, simplifying the harmonic-number closed form and verifying the recurrence numerically with symbolic tools.

Findings

The harmonic-number closed form simplifies the recurrence identity.

The proof confirms the conjectured recurrence for large n.

Symbolic verification supports the proof's validity.

Abstract

For the OEIS sequence A001711, contributed by N. J. A. Sloane long before the on-line era and identified there as the diagonal $T(n+4, 4)$ of a generalized-Stirling triangle, R. J. Mathar contributed in February 2020 the conjectured order-2 P-recursive recurrence \[ a(n) - (2n+5)\,a(n-1) + (n+2)^{2}\,a(n-2) \;=\; 0,\qquad n \ge 2. \] We give a one-page proof. Detlefs's harmonic-number closed form $a(n) = \tfrac{1}{4}(n+3)!\,(2 H_{n+3} - 3)$ collapses the left-hand side, after dividing through by $(n+1)!/4$, to a polynomial identity of $n$ with coefficient $H_{n+2}$. The harmonic-number coefficient simplifies to $(n+3) - (2n+5) + (n+2) = 0$ (using $H_{n+3} = H_{n+2} + \tfrac{1}{n+3}$ and $H_{n+1} = H_{n+2} - \tfrac{1}{n+2}$); the constant remainder is $-3 \cdot 0 = 0$ 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 $n = 2, \ldots, 5000$.