Example calculation(s) of Bu et al 2014 tensor products In Bu et al. 2014 they give a definition for a product on tensors as follows:

Let $\mathcal{A} \in \mathbb{C}^{n_1 \times n_2 \times \cdots \times n_2}$ and $\mathcal{B} \in \mathbb{C}^{n_1 \times n_2 \times \cdots \times n_{k+1}}$ be order $m \geq 2$ and $k \geq 1$ tensors, respectively. The product $\mathcal{A}\mathcal{B}$ is the following tensor $\mathcal{C}$ of order $(m-1)(k-1)+1$ with entries:
$$c_{i \alpha_1 \cdots \alpha_{m-1}} = \sum_{i_2,\cdots,i_m \in [n_2]} a_{ii_2 \cdots i_m}b_{i_2 \alpha_1} \cdots b_{i_m \alpha_{m-1}}$$
where $i \in [n_1]$, $\alpha_1, \cdots, \alpha_{m-1} \in [n_3] \times \cdots \times [n_{k+1}]$.

I am having difficulty understanding how to compute this operator. For one thing, I don't know why the tensor $\mathcal{A} \in \mathbb{C}^{n_1 \times n_2 \times \cdots \times n_2}$ ends with  $n_2$. This feels like it might be a typo, but I am not sure. But even that aside, I would appreciate an example calculation or two to understand how to compute this product.
 A: My interpretation of the paper does is define a product that trailing copies of a $n_2$-dimensional space of the first tensor with a single copy of space of this dimension of another tensor.
$$\begin{aligned}
\mathcal{A}∈ ℂ^{n_1×\overbrace{n_2×…×n_2}^{(m-1)-\text{times}}} = ℂ^{n_1}⊗(ℂ^{n_2})^{⊗(m-1)}
\end{aligned}$$
The big issue is the notation $\alpha_{1}, \ldots, \alpha_{m-1} \in\left[n_{3}\right] \times \cdots \times\left[n_{k+1}\right]$ which just makes it uncessarily hard to digest. We have $m-1$ multi-indices, each of which indexes into the tensor product $⨂_{i=3}^{k+1} ℂ^{n_{i}}$.
A imo better, more general and abstract way to say the same thing would be to say: We define a product (I'm shifting the indices $m→m+1$ and $k→k+1$)
$$\begin{alignedat}{2}
 \operatorname{prod}：&(U⊗V^{⊗m})×(V⊗W) {}⟶{} U⊗W^{⊗m}
\\ &(, ) {}⟼{} _{{\color{red}{}}, ({\color{green}{}}₁, …, {\color{green}{}}ₘ)} 
   = \Big(\sum_{{{\color{blue}{}}₁ … {\color{blue}{}}ₘ}}  _{{\color{red}{}},({\color{blue}{}}₁, {\color{blue}{}}₂, …, {\color{blue}{}}ₘ)} _{{\color{blue}{}}₁,{\color{green}{}}₁} _{{\color{blue}{}}₂,{\color{green}{}}₂}…_{{\color{blue}{}}ₘ,{\color{green}{}}ₘ}\Big)
\end{alignedat}$$
essentially by chaining $(U⊗V^{⊗m})×(V⊗W) ⟶ (U⊗V^{⊗m})×(V⊗W)^{⊗m} ⟶ U⊗W^{⊗m}$.
I.e. there is nothing unusual here, they simply make $m$-many copies of $$ via an outer product and then contract along the shared $V^{⊗m}$. Im case when $W = ⨂_{i=1}^{k} ℂ^{n_k}$, and $U=⨂_{j=1}^{l} ℂ^{p_l}$ this means that $$ is a tensor of order $m⋅k+l$.
