I am interested in a weighted version of the Catalan numbers. The generating function for this case, $$ f(x, y) = \sum_s \sum_n f_{s n} x^s y^n $$ (where the $y^n$ term is the weight), obeys the relation $$ f(x, y) = 1 + x y f(x, y) f(xy, y) $$
Main question:
Is there any hope of solving for $f(x, y)$ in closed form? Alternately, I'd be happy to get a closed form for the coefficients of the series expansion in $x$ of $f(x, y)$ at $x=0$ and a particular value of $y$, where $0 < y < 1$. An exact expression is not required; a good approximation will serve just as well.
Comments/thoughts:
-I haven't encountered any methods for solving such a problem before, so I don't know if there is a name for such problems. I apologize if the title and keywords are uninformative.
-Obviously when $y=1$, this reduces to the ordinary Catalan numbers.
-I believe that it may be possible to turn this into a coupled set of PDEs by defining $g(x, y) = f(xy, y)$. However, this seems to make the problem more complicated rather than less so.
-Since $f(xy, y)$ is known in terms of $f(x, y)$, we may be able to learn about the series at $x=0$ by examining the sequence $f(x y^m, y)$ as $m \to \infty$. However, I haven't been able to make this idea work in any concrete way yet.
Any ideas are appreciated. Thanks!
Edit: As Mark noted, this comes from trying to solve a recurrence relation of the form $$ f_{sn} = \sum_{s_1 + s_2 + 1 = s} \sum_{n_1 + n_2 + s_1 + 1 = n} f_{s_1 n_1} f_{s_2 n_2} $$ This arises when considering the first-return times in a one-dimensional discrete random walk where the probability of moving to the right decreases exponentially with position.