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

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 $q_0(z)G(z)+q_1(z)G^{\prime }(z)=R(z)$ 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.

arXiv comment12 pages