Recurrent relation with several indices ( How many $m$-dim cubes in $n$-dim cube )

Question

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.

Answer 1

This was an answer before the typo was edited by the OP on $Q_m^{n+1}=2Q_m^n+Q_{m-1}^{n-1}$.

Let $F(x,y)=\sum_{n,m\geq0}Q_n^mx^my^n$ be a generating function. Based on the recurrence relation alone, you should be getting $$F(x,y)=\frac{P(x,y)}{1-2y-xy^2};$$ for some polynomial $P(x,y)$ which depends on the initial conditions (this, I leave up to you).

Caveat. You last initial condition does not sync with the recurrence. For example, $Q(3,3)=2Q(3,2)+Q(2,1)=0$ but supposedly $Q(3,3)=1$ and $Q(3,2)=Q(2,1)=0$. A mismatch!

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After the edit by OP: $Q_m^{n+1}=2Q_m^n+Q_{m-1}^n$. Again, let $F(x,y)=\sum_{n,m\geq0}Q_n^mx^my^n$ be a generating function. Based on the recurrence relation alone, you should be getting $$F(x,y)=\frac{P(x,y)}{1-2y-xy}=P(x,y)\sum_{n=0}^{\infty}(2+x)^ny^n =P(x,y)\sum_{m,n\geq0}\binom{n}m2^{n-m}x^my^n;$$ for some polynomial $P(x,y)$ which depends on the initial conditions ($P=1$, I think). Therefore, $Q_m^n=\binom{n}m2^{n-m}$.