Let's consider a 2-variable generating function for the Dyck triangle numbers. Reccurence, satisfying to the triangle conditions is $d_{n, k}=d_{n-1, k-1}+d_{n-1, k}+d_{n-1, k-1}$, $d_{0, 0}=1$, $d_{0, j}=1$ where $d_{n, k}$ - is a number, placed in the $n$th row and in the $k$th column.
So, doing in the same way as for 1-variable case, we can evaluate $G(x, y)=\frac{x^{2}(1+y^{2})}{1-x}$ (i could have done some mistakes while evaluating it, but it does not affect the question's meaning at all).
How to get a closed form of $d_{n, k}$, if i know the exact form of $G(x, y)$?