  Magnitude is an invariant of metric spaces with origins in category theory.
Using potential theoretic methods, Barcel\'o and Carbery gave an algorithm for
calculating the magnitude of any odd dimensional ball in Euclidean space, and
they proved that it was a rational function of the radius of the ball. In this
paper an explicit formula is given for the magnitude of each odd dimensional
ball in terms of a ratio of Hankel determinants of reverse Bessel polynomials.
This is done by finding a distribution on the ball which solves the weight
equations. Using Schr\"oder paths and a continued fraction expansion for the
generating function of the reverse Bessel polynomials, combinatorial formulae
are given for the numerator and denominator of the magnitude of each odd
dimensional ball. These formulae are then used to prove facts about the
magnitude such as its asymptotic behaviour as the radius of the ball grows.
