Suppose I've got vectors $v = (1,-1)$ and $w = (1,1)$ and any $m \in \mathbb{N}$. Let $a = v \otimes v \otimes w^{\otimes m}$ and let $\tilde{a}$ be the sum over all $\binom{m}{2}$ unique vectors obtained by permuting the tensor coordinates of $a$. I'm interested in identifying the asymptotics of a function $f(m) = \|\tilde{a}\|_1$ where $\|\cdot\|_1$ is the usual $1$-norm in terms of $m$.

For example, when $m=0$, we simply have $\tilde{a} = v \otimes v$ and so $f(0) = 4$.

When $m=1$ we have $\tilde{a} = v \otimes v \otimes w + v \otimes w \otimes v + w \otimes v \otimes v$ and one can check that $f(1) = 12$.

When $m = 2$ we have $$\tilde{a} = v \otimes v \otimes w \otimes w + v \otimes w \otimes v \otimes w + v \otimes w \otimes w \otimes v\\+ w \otimes v \otimes v \otimes w + w \otimes v \otimes w \otimes v + w \otimes w\otimes v \otimes v$$

and one can check that $f(2) = 24$.

The problem seems simple, but I'm having a tough time finding a nice expression approximating $f(m)$. It seems related to some type of signed Chu-Vandermonde identity. Any insights or suggestions are appreciated. Thank you!