# On generalized Catalan numbers

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?

• Somebody has added many such "conjectures" to the OEIS. Most of them are probably 5-minutes exercises. – F. C. Apr 4 '17 at 18:56
• The generating function and the recurrence are "off by 1" in their index from the stated sequence, but indeed this can be algorithmically verified (for example with the gfun package of Maple). – Jay Pantone Apr 4 '17 at 19:02
• They are not "off by 1": note that the offset is given as $1$, so $a(1) = 2$, $a(2) = 1$, etc. – Robert Israel Apr 4 '17 at 19:07
• There is no good reason to call these numbers "generalized Catalan numbers". – Ira Gessel Apr 5 '17 at 3:31

## 1 Answer

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 $n \ge 7$.