Let $a(n)$ be A258173 i.e. sum over all Dyck paths of semilength $n$ of products over all peaks $p$ of $y_p$, where $y_p$ is the $y$-coordinate of peak $p$.
A Dyck path of semilength $n$ is a $(x,y)$-lattice path from $(0,0)$ to $(2n,0)$ that does not go below the $x$-axis and consists of steps $U=(1,1)$ and $D=(1,-1)$. A peak of a Dyck path is any lattice point visited between two consecutive steps $UD$.
The sequence begins with $$ 1, 1, 3, 12, 58, 321, 1975, 13265, 96073, 743753, 6113769, 53086314, 484861924, 4641853003 $$
Here $T(0)$ is a generating function for $a(n)$ where $$ T(k) = 1 - \cfrac{x}{(k + 2)x - \cfrac{1}{T(k+1)}} $$
Let $$ R(n, q) = R(n-1, q+1) + (q+1)\sum\limits_{j=0}^{q}R(n-1,j), \\ R(0, q) = [q = 0] $$
Here square bracket denotes Iverson bracket.
I conjecture that $$R(n,0)=a(n).$$
Here is the PARI/GP program to check it numerically:
a_upto(n) = my(CF = 1); for(i = 0, n, CF = 1 - x / ((n - i + 2) * x - 1 / CF) + x * O(x ^ n)); Vec(CF)
R_upto(n) = my(v1, v2, v3); v1 = vector(n + 1, i, i--; i == 0); v2 = v1; v3 = vector(n + 1, i, 0); v3[1] = 1; for(i = 1, n, for(q = 0, n - i, v2[q + 1] = v1[q + 2] + (q + 1) * sum(j = 0, q, v1[j + 1])); v1 = v2; v3[i + 1] = v1[1]); v3
test(n) = a_upto(n) == R_upto(n)
Is there a way to prove it?