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
$$
...+7x^5 + 12x^4y + 16x^3y^2 + 16x^2y^3 + 12xy^4 + 7y^5 + 5x^4 +\\\ +7x^3y + 9x^2y^2 + 7xy^3 + 5y^4 + 3x^3 + 4x^2y + 4xy^2+\\\ + 3y^3 + 2x^2 + 2xy + 2y^2 + x + y + 1
$$



A series representation
Your example should be 
$$ \prod\_{i=1}^\infty \prod\_{j=1}^i \dfrac{1}{1-x^{i-j} y^j} = 1+y+2{y}^{2}+xy+3{y}^{3}+2x{y}^{2}+{x}^{2}y+{x}^{3}y+3{x}^{2}{
y}^{2}+4x{y}^{3}+5{y}^{4} + \ldots$$
The triple product is
$$ \eqalign{\prod \_{i=1}^{\infty } & \prod \_{j=1}^{i} \prod \_{k=1}^{j}
 \dfrac{1}{ 1-{x}^{i-j}{y}^{j-k}{z}^{k}} \cr & =1+
z+2{z}^{2}+xz+yz+3{z}^{3}+2x{z}^{2}+2y{z}^{2}+{x}^{2}z+xyz+{y}
^{2}z\cr&+{x}^{3}z+3{x}^{2}{z}^{2}+{x}^{2}yz+4x{z}^{3}+x{y}^{2}z+3xy
{z}^{2}+{y}^{3}z+3{y}^{2}{z}^{2}+4y{z}^{3}+5{z}^{4}+\ldots\cr}
$$
EDIT: If you meant the $j$ and $k$ indexing to start from $0$ instead of $1$, 
$$
\eqalign{\prod \_{i=1}^{\infty } & \prod \_{j=0}^{i} \prod \_{k=0}^{j}
 \dfrac{1}{ 1-{x}^{i-j}{y}^{j-k}{z}^{k}} \cr & =
1+x+y+z+2\,{x}^{2}+2xy+2xz+2{y}^{2}+2yz+2{z}^{2}+3{x}^{3}+
4{x}^{2}y+4{x}^{2}z\cr&+4x{y}^{2}+5xyz+4x{z}^{2}+3{y}^{3}+4{
y}^{2}z+4y{z}^{2}+3{z}^{3}+5{x}^{4}+7{x}^{3}y+7{x}^{3}z+9{
x}^{2}{y}^{2}\cr&+11{x}^{2}yz+9{x}^{2}{z}^{2}+7x{y}^{3}+11x{y}^{2}
z+11xy{z}^{2}+7x{z}^{3}+5{y}^{4}+7{y}^{3}z\cr&+9{y}^{2}{z}^{2}+7
y{z}^{3}+5{z}^{4}+\ldots\cr}$$


