I'll use another way to give the question besides the title.


Define
$$ A\_k(n) = \frac{1}{2 \pi \mathrm{i}} \int\_{|q|=1} \left( \frac{1}{2 \pi \mathrm{i}}\int\_{|z|=1}\prod\_{j=-n}^{n}(1+qz^j) \frac{dz}{z} \right) \frac{1}{q^k} \frac{dq}{q}.$$


By some numeral calculation buy a simple program, I note that if we freeze $k$, then $A\_k(n)$ always satisfies a linear recurrence relations.


How to make a proof? Or anyone could give a counterexample? (A "appropriate" counterexample may be a sequence $A\_l(1), A\_l(2), \cdots, A\_l(m)$ for large enough $m$.)


Thanks for reading my problem. : )



A problem on counting k-subsets of {-n,-n+1,...,n-1,n} satisfying that sum of elements equal to 0
Here is a sketch of a proof. First I'll switch $q$ and $z$, since this is more consistent with standard “$q$-series” notation. 


Let $C\_k(n)$ be the coefficient of $z^k$ in $\prod\_{j=-n}^n (1+q^jz)$, so that $A\_k(n)$ is the constant term in $q$ in $C\_k(n)$.
By the $q$-binomial theorem
$$\prod\_{j=0}^{m-1} (1+q^iz) = \sum\_{k=0}^m {m \atopwithdelims[] k} q^{\binom k2}z^k,
$$
where 
$${m \atopwithdelims[] k} = \frac{(q^m-1)(q^{m-1}-1)\cdots (q^{m-k+1}-1)}{(q^k-1)\cdots (q-1)}.$$
It follows that
$$C\_k(n)={2n+1 \atopwithdelims[] k} q^{\binom k2 -kn}$$
Expanding the $q$-binomial coefficients, we see that 
$$C\_k(n)=\sum\_{l=-k}^k R\_l(q) q^{ln},$$
where each $R\_l(q)$ is a rational function of $q$.


Now let 
$$G\_k=\sum\_{n=0}^\infty C\_k(n) t^n.$$ 
Then 
$$G\_k=\sum\_{l}\frac{R\_l(q)}{1-tq^l}.$$
 Expanding $G\_k$ in partial fractions in $q$, we see that the constant term in $q$, which is $\sum\_n A\_k(n)t^n$, is a rational function of $t$, so $A\_k(n)$ satisfies a linear recurrence with constant coefficients.


For example,
$$G\_2 = \frac{(1+q)t}{(1-t)^2(1-q^2t)} + \frac{t(q+t)}{(1-t)^2(q^2-t)}.$$
The second term, when expanded as a power series in $t$, has only negative powers of $q$, so it contributes nothing to the constant term in $q$. The first term, when expanded as a power series in $t$, has only nonnegative powers of $q$, so we obtain the constant term in $q$ by setting $q=0$ in the first term and we find that 
$$\sum\_{n=0}^\infty A\_2(n) t^n = t/(1-t)^2 =\sum\_{n=0}^\infty nt^n.$$


