OFFSET
0,4
COMMENTS
The number of all noncrossing trees with n edges is given by A001764.
The number of nodes will be n + 1.
Rotational symmetry is only possible with an even number of nodes and with a rotation of 180 degrees (rotation by n/2 nodes). A tree with rotational symmetry will always include exactly one edge that connects diametrically opposite nodes.
The sequence satisfies a(2n) = A000139(2n)/2. - F. Chapoton, Sep 08 2023
LINKS
Andrew Howroyd, Table of n, a(n) for n = 0..200
EXAMPLE
Case n=3:
o---o o---o o---o o---o
| | \ \ /
o---o o o o---o o---o
In total there are 4 distinct noncrossing trees up to rotation.
MATHEMATICA
a[n_] := If[EvenQ[n], Binomial[3*n, n]/((n + 1)*(2*n + 1)), ((2*n + 1)*Binomial[(1/2)*(3*n - 1), (n - 1)/2] + Binomial[3*n, n]) / ((n + 1)*(2*n + 1))];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Dec 27 2017, after Andrew Howroyd *)
PROG
(PARI) a(n)={(binomial(3*n, n)/(2*n+1) + if(n%2, binomial((3*n-1)/2, (n-1)/2)))/(n+1)}
CROSSREFS
KEYWORD
nonn
AUTHOR
Andrew Howroyd, Dec 14 2017
STATUS
approved