A short proof of Mathar's 2021 recurrence conjecture for the Lehmer-Comtet diagonal A045406
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866914561464991744 |
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| author | Niu, Tong |
| author_facet | Niu, Tong |
| contents | 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 $n = 5, \ldots, 5000$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_12839 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A short proof of Mathar's 2021 recurrence conjecture for the Lehmer-Comtet diagonal A045406 Niu, Tong Combinatorics 05A15, 05A19, 11B37, 11B73, 11B83 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 $n = 5, \ldots, 5000$. |
| title | A short proof of Mathar's 2021 recurrence conjecture for the Lehmer-Comtet diagonal A045406 |
| topic | Combinatorics 05A15, 05A19, 11B37, 11B73, 11B83 |
| url | https://arxiv.org/abs/2605.12839 |