We have the following recurrence relation for $a_{n,m}$
$a_{n,m}=4a_{n+1,m-1}+\sum_{i=0}^{n-1}\sum_{j=0}^{m}a_{i,j}a_{n-1-i,m-j}$
with the boundary condition $a_{n,0}=c_n$ for $n\ge0$ where $c_n$ are the Catalan numbers and $a_{0,m}=0$ for $m>0$. After defining the generating function
$\phi(x,y)=\sum_{n,m=0}^\infty x^n y^m a_{n,m}$
the relation becomes an easy quadratic equation for $\phi(x,y)$ which basically solves the problem. This equation however involves $A(y)=\sum_{m=0}^\infty y^m a_{1,m}$ as a parameter, which is not given by the boundary condition. Anybody any ideas on how this $A(y)$ can be determined from the recursion rules without knowing the solution for $a_{n,m}$?