$\frac{{\left(q;q\right)}_{a_1+a_2+\cdots +a_n}}{{\left(q;q\right)}_{a_1}{\left(q;q\right)}_{a_2}\cdots {\left(q;q\right)}_{a_n}}=\text{coeff. of}{x}_1^0{x}_2^0\cdots x_n^0\text{in}{\prod }_{1\le j<k\le n}{\left(\frac{x_j}{x_k};q\right)}_{a_j}{\left(\frac{q{x}_k}{x_j};q\right)}_{a_k}$