A short proof of Mathar's 2016 recurrence conjecture for OEIS A176677

TL;DR

This paper provides a concise proof of Mathar's 2016 recurrence conjecture for OEIS sequence A176677, connecting the recurrence to an algebraic equation and an ODE for the generating function.

Contribution

The authors translate the convolution recurrence into an algebraic equation and verify Mathar's recurrence as a coefficient form of a linear ODE, offering a short proof.

Findings

The recurrence corresponds to an algebraic equation for the generating function.

Mathar's recurrence is derived from a linear inhomogeneous ODE.

The roots of the polynomial factorization match the singularities of the generating function.

Abstract

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.