I'll use a convenient way to give the question.

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 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. : )