Define a function $\varphi : \mathbb{N} \rightarrow \mathbb{N}$ as follows.
$$\varphi(n) = \sum_{i=1}^n \left[\binom{n}{i} \prod_{j=1}^i(i-j+1)^{2^j}\right]$$
The motivation is that according to this article, we have: for all natural $n$, $\varphi(n)$ is the cardinality of the free band on $n$ generators.
Questions.
I'd like to learn more about $\varphi$.
Q0. Does it have a standard name?
Q1. Are there other, apparently unrelated counting problems to which $\varphi$ is the solution?
Q2. Does $\varphi$ satisfy any interesting identities?
Q3. If so, is there a good resource for learning these identities?
Remark. This function helps explain where the number $14$ comes from in the Kuratowski closure-complement theorem. Basically, this happens because $$2(\varphi(2)+1) = 2(6+1) = 14.$$
To see the relevance of $\varphi(2),$ note that if $X$ is a topological space, then letting $k_X$ and $i_X$ denote the closure and interior operators on $X$ respectively, then the semigroup $S_X$ generated by $\{k_X,i_X\}$ is a band; furthermore, we can choose $X$ appropriately (e.g. $X = \mathbb{R}$) such that $S_X$ is the free band on these two elements.
