- Let $T(n,k)$ be A129179 (i.e., triangle read by rows: $T(n, k)$ is the number of Schroeder paths of semilength $n$ such that the area between the $x$-axis and the path is $k$ ($n \geqslant 0, 0 \leqslant k \leqslant n^2$)). Here generating function is $G(t,z)$ such that $$ G(t,z) = 1 + zG(t,z) + tzG(t,t^2z)G(t,z). $$
- Let $P(n,k)$ be a triangle read by rows such that $$ P(n, k) = \begin{cases} 0 & \textrm{if } k > n \\ 1 & \textrm{if } k = 0 \\ P(n-1, k) + z^{n-k}P(n + (n-k) \bmod 2, k-1) & \textrm{otherwise} \end{cases} $$
I conjecture that
$$ P(n+1, n+1) = \sum\limits_{k=0}^{n^2}T(n,k)z^k. $$
Is there a way to prove it?
