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

%I A296533 #22 May 02 2025 14:29:14
%S A296533 1,1,1,3,7,28,108,507,2431,12441,65169,351156,1926372,10746856,
%T A296533 60762760,347664603,2009690895,11723160835,68937782355,408323575275,
%U A296533 2434289046255,14598013278960,88011196469040,533216762488020,3245004785069892,19829769013792908
%N A296533 Number of nonequivalent noncrossing trees with n edges up to rotation and reflection.
%C A296533 The number of all noncrossing trees with n edges is given by A001764.
%C A296533 The number of nodes will be n + 1.
%H A296533 Andrew Howroyd, <a href="/A296533/b296533.txt">Table of n, a(n) for n = 0..200</a>
%H A296533 Bernold Fiedler, <a href="https://arxiv.org/abs/2504.20503">Real-time blow-up and connection graphs of rational vector fields on the Riemann sphere</a>, arXiv:2504.20503 [math.DS], 2025. See p. 24.
%H A296533 Marc Noy, <a href="https://dx.doi.org/10.1016/S0012-365X(97)00121-0">Enumeration of noncrossing trees on a circle</a>, Discrete Math., 180, 301-313, 1998.
%F A296533 a(2n) = (A296532(2n) + A001764(n))/2, a(2n-1) = (A296532(2n-1) + A006013(n-1))/2.
%F A296533 a(2n) = A005034(2n).
%e A296533 Case n=3:
%e A296533    o---o   o---o   o---o
%e A296533    |       | \       \
%e A296533    o---o   o   o   o---o
%e A296533 In total there are 3 distinct noncrossing trees up to rotation and reflection.
%t A296533 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));
%t A296533 Table[a[n], {n, 0, 25}] (* _Jean-François Alcover_, Dec 27 2017, after _Andrew Howroyd_ *)
%o A296533 (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))}
%Y A296533 Cf. A001764, A005034, A006013, A296532 (up to rotation only).
%K A296533 nonn
%O A296533 0,4
%A A296533 _Andrew Howroyd_, Dec 14 2017