Given two operators $A$ and $B$ that do not commute and an natural number $k$, I would like to express in a concise and clear manner the sum of all operators that are products of $A$'s and $B$'s and are such that the number of $A$'s plus twice the number of $B$'s is equal to $k$. For example, in the case where $k = 4$, all the terms in the sum are: $$AAAA, \, AAB, \, ABA, \, BAA, \,BB. $$ At the moment I use: \begin{equation} \sum_{i=0}^{\lfloor k / 2 \rfloor} \, \sum_{v \in I_i} \, \prod_{j=1}^{k-i} \left( v_i \, A + (1 - v_i) \, B \right), \text{ where } I_i = \{v \in \{0, 1\}^{k-i}: \sum v_j = i\}, \end{equation} but it looks overly complicated. ($i$ is the number of times that $B$ appears in the product.)
EDIT: In fact, the operator I am looking for is at the position 1, 1 of the matrix: \begin{equation} \begin{pmatrix} A & B \\ I & 0 \end{pmatrix}^k. \end{equation}
You can define the free product set of $A$ and $B$ as $\mathcal{P} = \{ c_1 \cdots c_p \mid c_j \in \{ A, B \}, \ p = 0, 1, 2, \ldots \}$ and $\operatorname{deg} : \mathcal{P} \to \mathbb{N}_0$ by $\operatorname{deg}(A) = 1,$ $\operatorname{deg}(B) = 2,$ and $\operatorname{deg}(xy) = \operatorname{deg}(x) + \operatorname{deg}(y).$ Then you can write the sum as $$\sum \{ x \in \mathcal{P} \mid \operatorname{deg}(x) = k \}.$$