Recall the construction of the tangent bundle: we write $$TM = \bigsqcup_{p \in M}T_p M$$ and define it as the prevector bundle with local trivializations $[\gamma] \mapsto (\gamma(0), (x\gamma)'(0))$ for $x$ a chart about the point $p = \gamma(0)$.
Let $T_l^k(V)$ be the space of all $\binom{k}{l}$-tensors on $V$, which we will look at as multilinear maps $$V^* \times \cdots \times V^* \times V \times \cdots \times V \to \mathbb{R}$$ where there are $l$ copies of $V^*$ and $k$ copies of $V$. In similar fashion we define the bundle of $\binom{k}{l}$-tensors as $$T_l^k(M) = \bigsqcup_{p \in M}T_l^k(T_p M).$$ The proposed local trivialization is $\pi^{-1}(U) \to U \times \mathbb{R}^{n^{k + l}}$ where $\pi$ is the canonical projection onto $M$ and the map is given by mapping $$F \colon T_p^* M \times \cdots \times T_p^* M \times T_p M \times \cdots \times T_p M \to \mathbb{R}$$
to $$(p, F_{i_1, \dots, i_k}^{j_1, \dots, j_l}).$$ The latter notation I am not sure I understand. The only way I can make sense of this (for dimensional reasons) is that $F_{i_1, \dots, i_k}^{j_1, \dots, j_l}$ is all different combinations $F(\omega^{j_1}, \dots, \omega^{j_l}, X_{i_1}, \dots, X_{i_k})$ where the entries are respective basis elements of $T^*_p(M)$ and $T_p M$.
Is this correct or have I misunderstood the notation?