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. https://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.

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EDIT: Here are is the formula written out. 

For $m\in M$, the mean curvature at the point $\exp_m(ru)$ (for $r >0$ and $u \in T_mM$ a unit vector) is given by
$$
\frac{n-1}{r} + \alpha_1r+\alpha_2r^2+O(r^3)
$$
where
\begin{align*}
\alpha_1 & = -\frac13 Ric|_m(u,u)\\
\alpha_2 & = -\frac14(\nabla Ric)|_m (u,u,u).
\end{align*}