How to find the end of a series representation of the product $$ \prod_{\substack{i=1...\infty\\\ j=0...i\\\ k=0...j}}\frac{1}{1-x^{i-j}y^{j-k}z^{k}}? $$

For example for product $$ \prod_{\substack{i=1...\infty\\\ j=0...i}}\frac{1}{1-x^{i-j}y^j} $$ the ends of series is $$ ...+7x^5 + 12x^4y + 16x^3y^2 + 16x^2y^3 + 12xy^4 + 7y^5 + 5x^4 +\\\ +7x^3y + 9x^2y^2 + 7xy^3 + 5y^4 + 3x^3 + 4x^2y + 4xy^2+\\\ + 3y^3 + 2x^2 + 2xy + 2y^2 + x + y + 1 $$