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

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 $a(n)=\frac{1}{4}(n+3)!(2H_{n+3}-3)$ 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 $H_{n+3}=H_{n+2}+\frac{1}{n+3}$ and $H_{n+1}=H_{n+2}-\frac{1}{n+2}$); 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 $n=2,\dots ,5000$.

arXiv comment8 pages