Higher order terms in Riemann's formula (normal coordinates)

Question

In normal coordinates, we have $$ \Gamma_{ijk}(x)=-\frac{1}{3}(R_{ijkl}(0)+R_{ikjl}(0))x^{l}+O(x^2) $$ where $ \Gamma_{ijk}\equiv g_{is}\Gamma^s_{jk}$. My question is whether the higher order terms can be expressed solely in terms of $R_{ijkl}$ and its partial derivatives (evaluated at zero of course).

Answer 1

The answer would appear to be yes. In fact, the expansion can be derived from the results of this paper which gives a closed form expression for the metric in terms of the curvature in normal coordinates.