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?

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 $n \ge 7$.

`gfun`

package of Maple). $\endgroup$ – Jay Pantone Apr 4 '17 at 19:02