A296533 Number of nonequivalent noncrossing trees with n edges up to rotation and reflection.

1, 1, 1, 3, 7, 28, 108, 507, 2431, 12441, 65169, 351156, 1926372, 10746856, 60762760, 347664603, 2009690895, 11723160835, 68937782355, 408323575275, 2434289046255, 14598013278960, 88011196469040, 533216762488020, 3245004785069892, 19829769013792908

Offset: 0

Author

Andrew Howroyd, Dec 14 2017

Keywords

• nonn

Comments

The number of all noncrossing trees with n edges is given by A001764.

The number of nodes will be n + 1.

Examples

			Case n=3:
   o---o   o---o   o---o
   |       | \       \
   o---o   o   o   o---o
In total there are 3 distinct noncrossing trees up to rotation and reflection.
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..200
• Bernold Fiedler, Real-time blow-up and connection graphs of rational vector fields on the Riemann sphere, arXiv:2504.20503 [math.DS], 2025. See p. 24.
• Marc Noy, Enumeration of noncrossing trees on a circle, Discrete Math., 180, 301-313, 1998.

Crossrefs

Cf. A001764, A005034, A006013, A296532 (up to rotation only).

Programs

Mathematica

a[n_] := (If[OddQ[n], 3*Binomial[(1/2)*(3*n - 1), (n - 1)/2], Binomial[3*n/2, n/2]] + Binomial[3*n, n]/(2*n + 1))/(2*(n + 1));
Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Dec 27 2017, after Andrew Howroyd *)

PARI

a(n)={(binomial(3*n, n)/(2*n+1) + if(n%2, 3*binomial((3*n-1)/2, (n-1)/2),  binomial(3*n/2, n/2)))/(2*(n+1))}

Formula

a(2n) = (A296532(2n) + A001764(n))/2, a(2n-1) = (A296532(2n-1) + A006013(n-1))/2.

a(2n) = A005034(2n).