How to think of $k$th covariant derivatives in coodinates

Question

On a Riemannian manifold $(M, g)$ one can simply define the $C^k$ norm of a tensor $T$ as the norm of $\nabla \ldots \nabla T $ with respect to $g$. I am trying to think of $\nabla^{k} T$ in terms of a normal coordinate system. Suppose that $T = T^{i_1, \ldots, i_n}_{j1, \ldots, j_m} \frac{\partial }{\partial x_{i_1}} \otimes \ldots \otimes \frac{\partial }{\partial x_{i_n}} \otimes dx^{j_1} \otimes dx^{j_m}$. what should the $k$th covariant derivative of it look like in a normal local coordinate system?