Say we are given an oriented and complete Riemannian manifold $(M,g)$ such that for a compact set $K \subset M$ we have that $ M \setminus K $ is diffeomorphic to the complement of the closed ball $\mathbb{R}^n \setminus \bar{B}_{1}(0) $ and furthermore, in the standart coordinates $x_1,...,x_n$ of $\mathbb{R}^n$ given by this diffeomorphism, the metric looks like $$ g_{ij} = \delta_{ij} + a_{ij}, $$ where $a_{ij} = O(1/r)$, $\partial_k a_{ij} = O(1/r^2)$ and $\partial_l\partial_k a_{ij} = O(1/r^3)$ (essentially, we are supposing $(M,g)$ to be asymptotically flat).
Will it then be true that the Christoffel symbols of first and second order satisfy the following equality? $$ \Gamma_{ij}^k = \Gamma_{kij} + O(1/r) $$