What I want is to count the number of binary trees on $n$ nodes, except when a node has only one child, I don't distinguish between left and right. So let $T_n$ be the number of such trees on $n$ nodes, we have
\begin{align*} T_n=\left(\sum_{k=0}^{n-1}T_k\cdot T_{n-k-1}\right)-T_{n-1}. \end{align*}
I am not sure how to solve this recurrence. If we proceed the same way as in counting normal binary trees (where left and right matters when there's a single child) using generating function $g(s)=\sum_{n=0}^\infty T_ns^n$ and squaring it, I get that
\begin{align*} g(s)=\frac{1+s-\sqrt{1-2s-3s^2}}{2s}. \end{align*}
Calculating the series expansion around zero seems to give me what I want, but I cannot get this expression in the form of $\sum_nT_ns^n$ so the quantity $T_n$ comes out...