The Number of [Old-Time] Basketball Games with Final Score $n$:$n$ where the Home Team was Never Losing but also Never Ahead by More Than $w$ Points

Abstract

We show that the generating function (in $n$) for the number of walks on the square lattice with steps $(1,1), (1,-1), (2,2)$ and $(2,-2)$ from $(0,0)$ to $(2n,0)$ in the region $0 \leq y \leq w$ satisfies a very special fifth order nonlinear recurrence relation in $w$ that implies both its numerator and denominator satisfy a linear recurrence relation.