Enumeration of lattice paths of a specific type 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
 A: It seems the first solution to this problem appeared in Theorem 4 of
Raschel, Kilian, Counting walks in a quadrant: a unified approach via boundary value problems, J. Eur. Math. Soc. (JEMS) 14, No. 3, 749-777 (2012). ZBL1238.05014.
Notice that the walk corresponds to the fourth singular walk of Figure 2. The kernel $K(x,y;z)$ of the walk is given by $$K(x,y;z) = x y z(x +x/y +y +y/x -1/z).$$
Let $X_0(y) = X_0(y;z)$ (resp. $Y_0(x)=Y_0(x;z)$) be one of the solutions $x$ (resp. $y$) to $K(x,y;z) = 0$. Then Theorem 4 states that
$$Q(x,y;z) = \sum_{m,n\geq 1} x^{m-1}y^{n-1} \sum_{\ell\geq 0} z^l a_{m,n}(\ell) $$
satisfies
$$ Q(x,y;z) = \frac{1}{K(x,y;z)}\left(z x^2Q(x,0;z)+z y^2 Q(0,y;z)-xy\right)$$
with
$$ Q(x,0;z) = \frac{1}{zx^2} \sum_{p\geq 0} Y_0 \circ (X_0\circ Y_0)^{\circ p}(x;z)\,\left[ (X_0\circ Y_0)^{\circ p}(x;z) - (X_0\circ Y_0)^{\circ (p+1)}(x;z) \right].$$
Here $f^{\circ p}$ means $f\circ \cdots \circ f$ with $p$ occurrences of $f$. Notice that by symmetry $Q(0,y;z) = Q(y,0;z)$.
A quick check with Mathematica reproduces the table for $\ell=7$:
k = x y z (x + x/y + y + y/x - 1/z);
x0[y_] = x /. Solve[k == 0, x][[1]] // FullSimplify[#, y > 0 && x > 0] &;
y0[x_] = y /. Solve[k == 0, y][[1]] // FullSimplify[#, y > 0 && x > 0] &;
Series[SeriesCoefficient[1/k (Sum[y0[xp] (xp - x0[y0[xp]]) /. 
   xp -> Nest[x0[y0[#]] &, x, p], {p, 0, 2}] 
   // -x y + # + (# /. x -> y) &), {z, 0, 7}], {x, 0, 7}, {y, 0, 7}]

yields the output
$$
\begin{aligned}
&\ \ \ \left(y+21 y^2+80 y^3+125 y^4+85 y^5+21 y^6+y^7+O\left(y^8\right)\right)\\+&x\ 
   \left(1+28 y+139 y^2+254 y^3+210 y^4+76 y^5+7 y^6+O\left(y^8\right)\right)\\+&x^2
   \left(21+139 y+306 y^2+308 y^3+140 y^4+21 y^5+O\left(y^8\right)\right)\\+&x^3
   \left(80+254 y+308 y^2+168 y^3+35 y^4+O\left(y^8\right)\right)\\+&x^4
   \left(125+210 y+140 y^2+35 y^3+O\left(y^8\right)\right)\\+&x^5 \left(85+76 y+21
   y^2+O\left(y^8\right)\right)\\+&x^6 \left(21+7
   y+O\left(y^8\right)\right)\\+&x^7+O\left(x^8\right)
\end{aligned}
$$
A: This is an extension of the accepted answer. Using it, and ideas from some other papers, I managed to obtain an alternative expression for the generating function which I find interesting.
As explained in that answer, the problem reduces to finding $Q(x,0,z):=\sum_{i,\ell}a_{n,1}(\ell)x^nz^\ell$.
Let $z=\frac1{q+q^{-1}}$ and $x=-\frac{(1-q) (1-q^2) t}{(1-t) (1-q t)q}$. Then,
$$
x^2zQ(x,0,z)=-x+\frac{1-4z^2}{z^2}\sum _{k=1}^\infty \frac{(-1)^kq^kt}{\left(1-q^kt\right)^2}.
$$
The series $\sum _{k=1}^\infty \frac{(-1)^kq^kt}{\left(1-q^kt\right)^2}$ seems to be related to elliptic functions, Jacobi forms and such stuff, but I cannot figure out how exactly. Maybe I will ask a separate question about it.
Few words about the proof. The expression for the generating function in the other answer can be equivalently described as follows. Suppose $x$, $y$ satisfy $(x^2+y^2+x^2y+xy^2)z=xy$. Then,
$$
(*)\qquad\qquad\qquad\qquad\qquad\qquad x^2zQ(x,0,z)=xy-y^2zQ(y,0,z).\qquad\qquad\qquad\qquad\qquad\qquad
$$
Now the curve $(x^2+y^2+x^2y+xy^2)z=xy$ admits rational parametrizations $x(t)$, $y(t)$; moreover it is possible to choose such parametrization that satisfies $y(t)=x(qt)$, with $z=\frac1{q+q^{-1}}$ and $x(t)=-\frac{(1-q) (1-q^2) t}{(1-t) (1-q t)q}$. Then, denoting $F(t,q)=\frac{x(t)^2}{q+q^{-1}}Q\left(x(t),0,\frac1{q+q^{-1}}\right)$, we get from $(*)$ above the following functional equation for $F$:
$$
F(t,q)=\frac{(1-q)^2 \left(1-q^2\right)^2 t^2}{(1-t) (1-q t)^2 \left(1-q^2 t\right)q}-F(qt,q).
$$
This leads more or less straightforwardly to the claimed equality.
Update
Expression in terms of "original" variables can be also derived from this. Although less pretty, it is, I believe, still more explicit than the one in terms of iterated radicals.
We have
$$
x^2zQ(x,0,z)=-x+\frac{1-4z^2}{z^2}\sum _{k=1}^\infty\frac{(-1)^k}{A(x,z)T_k\left(\frac{1}{2 z}\right)+B(x,z)U_{k-1}\left(\frac{1}{2 z}\right)-2},
$$
where $T$ and $U$ are the Chebyshev polynomials, while $A(x,z)$ and $B(x,z)$ are quadratic irrationalities determined by
$$
\begin{aligned}
A+\frac{2z}{1+2z}B&=2\left(1-\frac{1-2 z}{x z}\right),\\
A^2-\frac{4z^2}{1-4z^2}B^2&=4
\end{aligned}
$$
A: Bousquet-Melou's work might be pertinent. See 
https://arxiv.org/abs/1708.06192
Figure 4 there seems to be your lattice walk.
