A Riemannian metric $g$, in normal coordinates, has a Taylor expansion $$g_{ij}(x) \ = \ \delta_{ij} \ - \ \frac{1}{3}R_{iklj}x^kx^l \ - \ \frac{1}{3}R_{iklj;m}x^kx^lx^m \ + \ \frac{1}{180}\left(8R_{ilmk}R_{jpqk}-9R_{ilmj;pq}\right)x^kx^lx^px^q $$ $$\ + \ \frac{1}{90}\left(4R_{iklr;m}R_{jpqr}-R_{iklj;mpq}\right)x^kx^lx^mx^px^q\ + \ \cdots $$ This is proven here, and the proof was asked for earlier. Each term in the series is a product of $R$ and its covariant dervatives. $$\text{}$$ This series is hideous, and I would like to know if there is a cleaner or more symmetric way of expressing the coefficients. If not, can we at least say something about them which makes them seem more natural?
For instance, if $f$ is a function, its Taylor series $$f(x) \ = \ f(0) \ + \ f_i(0)x^i \ + \ \frac{1}{2!}f_{ij}(0)x^ix^j \ + \ \cdots$$ is pretty easy to remember, and its symmetries are that the coefficients are symmetric in the indices $i,j,...$. Hopefully $f$ being matrix-valued doesn't mean its Taylor series is suddenly a structureless mess, but there are even more symmetries, though they might be harder to see.
