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 $\binom{n}k\_q=\frac{(q)\_n}{(q)\_k(q)\_{n-k}}$.


(0) It's almost silly, but the sum of elementary functions of the specialization $\pmb{q}=(q,q^2,\dots,q^n)$:
$$e\_0(\pmb{q})+e\_1(\pmb{q})+\cdots+e\_n(\pmb{q})=\prod\_{j=1}^n(1+q^j).$$


(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 (do you?) but 
$$\sum\_{k=0}^nq^k\binom{n}k\_{q^2}=\prod\_{j=1}^n(1+q^j).$$


(3) The $H$-polynomial of a symplectic monoid $MSp\_n$ ([see this paper, page 13](https://arxiv.org/pdf/0812.3105v1.pdf)): 
$$\sum\_{k=0}^n(-1)^kq^{k^2}\binom{n}k\_{q^2}^2\prod\_{i=1}^k(1-q^{2i})\prod\_{j=1}^{n-k}(1+q^j)^2=\prod\_{j=1}^{2n}(1+q^j),$$
although the authors did not seem to be aware of the RHS.



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


Thank you.



what else is in $\prod_{j=1}^n(1+q^j)$?
Up to scaling, $\prod\_{j=1}^n(1+q^j)$ is the character of the principal specialization of
the spinor representation of $\mathfrak{so}(2n+1)$. This was first
explicitly stated by J. W. B. Hughes, Lie algebraic proofs of some theorems on partitions, in *Number Theory and
Algebra*, Academic Press, 1977, pp. 135--155.


