See http://link.springer.com/article/10.1007%2FBF02395060 Lemma 12.2 (the $\gamma_i$ terms are defined on page 167, also see $\S2$ for the curvature notations). 

An alternative "do-it yourself" approach to what you want might be to consider the expansion of the metric in normal coordinates, e.g. http://mathoverflow.net/questions/185527/riemanns-formula-for-the-metric-in-a-normal-neighborhood, which is not that hard to prove. Then plug this into the coordinate expression for the Laplacian and find a power series for the Laplacian. Changing into polar coordinates should then presumably give you what you want.