I'm giving context although is not necessary in order to answer the question (I hope)
A Schröder path is a Dyck path where you can also move East, i.e $+(2,0)$. Call the set of Schröder paths of lenght $2n$ $Sch_n$. Here we are trying to count those adding a catalitic variable $y$, keeping track the number of the steps in South and East after the last Nord. In the Kernel method for Schröder paths, i.e trying to find an equation for $f(x,y) = \sum\limits_{n,k} f_{n,k}x^ny^k$, I reduced to $$(1-xy+\frac{xy^2}{1-y})f(x,y) = y+\frac{xy^2}{1-y}f(x,1)$$
An $y_0(x)$ which "kills" LHS is $$y_0(x) = \frac{1+x-\sqrt{1-6x+x^2}}{4x}$$ So $y \to y_0(x)$ gives $f(x,1) = -\frac{1-y_0(x)}{xy_0(x)}$. According to writer notes this should bne $$f(x,1) = \frac{1-x-\sqrt{1-6x+x^2}}{2x}$$
Anyone seeing this? Any help or hint would be appreciated.
