Riemannian Distance function taylor series

Question

Given a Riemannian manifold $ (M,g) $, in a neighborhood of a fixed point $ x $, the distance function $ d^2(x,\cdot) $ is smooth. In coordinates, the taylor series starts off like: $$ d^2(x,y) = g_{ij} (x-y)^i (x-y)^j + c_{ijk} (x-y)^i (x-y)^j (x-y)^k + \cdots $$ What are the $ c_{ijk} $? I am pretty sure they are related to the curvature, but I haven't been able to find a quick answer.