One of the approaches to "Special" meanders led (in particular) to the following question:

What is the number $a_{m,n}(\ell)$ of $\ell$-step paths from $(1,1)$ to $(m,n)$ using the following four kinds of steps: $(i,j)\mapsto$ $(i+1,j)$, $(i+1,j-1)$, $(i,j+1)$ or $(i-1,j+1)$ and with the restriction that $i$ and $j$ stay positive at each step?

The only thing I managed to find out is sort of a trivial reformulation of the definition: the polynomials $$ P_\ell(x,y):=\sum_{m,n\geqslant1}a_{m,n}(\ell)x^my^n $$ satisfy a recurrence: since $P_\ell(x,0)=P_\ell(0,y)=0$, each $F_\ell(x,y):=(x+y+x/y+y/x)P_\ell(x,y)$ is a polynomial, and we have $$ P_{\ell+1}(x,y)=F_\ell(x,y)-F_\ell(x,0)-F_\ell(0,y). $$

In a way of example - here is the table of the $a_{m,n}(7)$: $$ \begin{array}{cccccccc} 0 & 1 & 21 & 80 & 125 & 85 & 21 & 1 \\ 1 & 28 & 139 & 254 & 210 & 76 & 7 & 0 \\ 21 & 139 & 306 & 308 & 140 & 21 & 0 & 0 \\ 80 & 254 & 308 & 168 & 35 & 0 & 0 & 0 \\ 125 & 210 & 140 & 35 & 0 & 0 & 0 & 0 \\ 85 & 76 & 21 & 0 & 0 & 0 & 0 & 0 \\ 21 & 7 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \end{array} $$

As literature on the subject of lattice paths is way too vast for me to handle, this is mostly a reference request: I believe that working hardly enough I could find at least a generating function, but I also believe it must be done somewhere already.

In fact a more (or less?) specific question in this direction is whether there is some kind of searchable database of combinatorial objects where I could find such things. There is a question Combinatorial Databases here on MO but I could not find anything about path enumeration in the answers. The closest I could get is the Dyck path enumeration on FindStat