The signature of a sample path is a formal series of iterated integrals along the path. The expected signature of a stochastic process gives a summary of the process that is especially useful for studying stochastic differential equations driven by the process. Lyons-Ni derived a partial differential equation for the expected signature of Brownian motion, starting at a point z in a bounded domain, until it hits the boundary of the domain. We focus on the domain of planar unit disc centred at 0. Motivated by recently found explicit formulae for the expected hyperbolic development of this process in terms of Bessel functions, we derive a tensor series representation for this expected signature, coming from studying Lyons-Ni’s PDE. Although the representation is rather involved, it simplifies significantly to give a formula for the polynomial leading degree term in each tensor component of the expected signature.