A short proof of Mathar's 2016 recurrence conjecture for OEIS A176677
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arXiv
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2026
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| _version_ | 1866917463269048320 |
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| author | Niu, Tong |
| author_facet | Niu, Tong |
| contents | For the OEIS sequence A176677, defined by the quadratic convolution recurrence $a(0) = a(1) = 1$ and $a(n+1) = \sum_{p=0}^n a(p) a(n-p) - 1$ for $n \ge 1$, R.~J.~Mathar contributed in March 2016 the conjectured order-4 P-recursive recurrence \[ (n+1)\,a(n) + 2(-3n+1)\,a(n-1) + (9n-13)\,a(n-2) - 4\,a(n-3) + 4(-n+4)\,a(n-4) = 0. \] We give a short proof. The convolution recurrence translates directly into the algebraic equation $z(1-z) G(z)^2 - (1-z) G(z) + (1 - z - z^2) = 0$ for the ordinary generating function $G(z)$, and Mathar's recurrence then drops out as the coefficient form of a 1st-order linear inhomogeneous ODE $q_0(z) G(z) + q_1(z) G'(z) = R(z)$ that we verify by polynomial division modulo the algebraic equation. The polynomial $q_1(z)$ admits the factorization $q_1(z) = -z(z-1)(2z-1)(2z^2 + 3z - 1)$, whose roots are exactly the singularities of $G$. Deutsch's combinatorial interpretation (Motzkin paths of length $n-1$ with two-coloured level-zero horizontal steps) is preserved. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_04369 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A short proof of Mathar's 2016 recurrence conjecture for OEIS A176677 Niu, Tong Combinatorics 05A15, 05A19, 11B37, 33F10 For the OEIS sequence A176677, defined by the quadratic convolution recurrence $a(0) = a(1) = 1$ and $a(n+1) = \sum_{p=0}^n a(p) a(n-p) - 1$ for $n \ge 1$, R.~J.~Mathar contributed in March 2016 the conjectured order-4 P-recursive recurrence \[ (n+1)\,a(n) + 2(-3n+1)\,a(n-1) + (9n-13)\,a(n-2) - 4\,a(n-3) + 4(-n+4)\,a(n-4) = 0. \] We give a short proof. The convolution recurrence translates directly into the algebraic equation $z(1-z) G(z)^2 - (1-z) G(z) + (1 - z - z^2) = 0$ for the ordinary generating function $G(z)$, and Mathar's recurrence then drops out as the coefficient form of a 1st-order linear inhomogeneous ODE $q_0(z) G(z) + q_1(z) G'(z) = R(z)$ that we verify by polynomial division modulo the algebraic equation. The polynomial $q_1(z)$ admits the factorization $q_1(z) = -z(z-1)(2z-1)(2z^2 + 3z - 1)$, whose roots are exactly the singularities of $G$. Deutsch's combinatorial interpretation (Motzkin paths of length $n-1$ with two-coloured level-zero horizontal steps) is preserved. |
| title | A short proof of Mathar's 2016 recurrence conjecture for OEIS A176677 |
| topic | Combinatorics 05A15, 05A19, 11B37, 33F10 |
| url | https://arxiv.org/abs/2605.04369 |