Actually, I will be asking two, but related, questions.

> **Question 1.** Is there some theory about recurrent relations with *several* indices. For example, If we have a relation on $A_{i;j}$: $$A_{i+\alpha; j+\beta} = F(A_{i+\alpha-1;j+\beta -1}, \ldots, A_{i;j}).$$

And related

> **Question 2.** Let $m, n$, $Q_{m}^ {n}\in \mathbb{N}$. Then solve given 2-recurrent relation:
$$Q_{m}^{n+1} = 2Q_{m}^{n} + Q_{m-1}^{n}$$
with the following conditions on $Q$:
\begin{align}
Q_{j}^{j} &= 1,\\
Q_{k}^{l} &= 0,\quad \text{for } k>l.
\end{align}


I have solved the last problem by pattern recognition followed by induction , two ways that I do not like to prove something. I would like to get more rough method.  
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This relation naturally appeared when I tried to count the number of $m$-dim edges in $n$-dim cube (vertex is $0$-dim, edge - 1-dim. square - $2$-dim and etc). In this context the answer is obvious I think. I will post my solution below.