From time to time, I run into the finite product $\prod_{j=1}^n(1+q^j)$. And, the more it happens the more fascinated I've become. So, herein, I wish to get help in collecting such results. To give some perspective into what I look for, check out the below examples. First, some nomenclature: $(q)_k=(1-q)(1-q^2)\cdots(1-q^k)$ and $\binom{n}k_q=\frac{(q)_n}{(q)_k(q)_{n-k}}$.

(1) The classical $q$-binomial theorem, which results from counting restricted distinct partitions or number of weighted tilings: $$\sum_{k=0}^nq^{\binom{k+1}2}\binom{n}k_q=\prod_{j=1}^n(1+q^j).$$

(2) I can't remember where I saw this (help?) but it might be Stanley's interpretation on upper-triangular matrices with entries in $\mathbb{F}_q$ where tapping rows is allowed: $$\sum_{k=0}^nq^k\binom{n}k_{q^2}=\prod_{j=1}^n(1+q^j).$$

(3) The $H$-polynomial of a symplectic monoid $MSp_n$ (see this paper, page 13): $$\sum_{k=0}^n(-1)^kq^{k^2}\binom{n}k_{q^2}^2\prod_{i=1}^k(1-q^{2i})\prod_{j=1}^{n-k}(1+q^j)^2=\prod_{j=1}^{2n}(1+q^j).$$

QUESTION. Can you provide such formulas (from any field) with the same RHS (always finite product) as in above, together with resources or references?

Thank you.