Counting some things in homological algebra, I found this sequence:
<https://oeis.org/A025242>.
Is there a good motivation why this sequence is called "generalized Catalan numbers"?
In the link there can be found the conjecture that the sequences satisfies $n\*(n+1)\*a(n) +(n^2+n+2)\*a(n-1) +2\*(-9\*n^2+15\*n+17)\*a(n-2) +2\*(5\*n+4)\*(n-4)\*a(n-3) +(n+1)\*(n-6)\*a(n-4) +(5\*n+4)\*(n-7)\*a(n-5)=0$. Im a bit surprised that this is a conjecture, since the generating function is explicitly known. Is the conjecture still open?



On generalized Catalan numbers
The g.f. satisfies the differential equation 
$$\eqalign{&\left( 4+22\,x-38\,{x}^{2}-10\,{x}^{3}-58\,{x}^{4} \right) y \left( x
 \right)\cr + &\left( 2+4\,x-60\,{x}^{2}+38\,{x}^{3}+4\,{x}^{4}+24\,{x}^{5
} \right) {\frac {\rm d}{{\rm d}x}}y \left( x \right)\cr +& \left( x+{x}^{
2}-18\,{x}^{3}+10\,{x}^{4}+{x}^{5}+5\,{x}^{6} \right) {\frac {{\rm d}^
{2}}{{\rm d}{x}^{2}}}y \left( x \right)\cr = &-34\,{x}^{5}-6\,{x}^{4}-72\,{
x}^{3}-50\,{x}^{2}+22\,x+4
}
$$
which implies the conjectured recurrence for $n \ge 7$.


