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%I #11 Feb 17 2019 19:34:09
%S 0,0,0,0,0,4,14,60,210,728,2442,8252,27716,93924,319964,1098900,
%T 3800928,13244836,46460738,164015272,582353976,2078812492
%N Number of chiral non-crossing partition patterns of n points on a circle, divided by 2.
%C Half of the number of those rotation-inequivalent patterns of non-crossing partitions of n (equally spaced) points on a circle which are not invariant under reflections. Division by two counts one pattern from each chiral (Right-handed,Left-handed) pair.
%H D. Callan and L. Smiley, <a href="https://arxiv.org/abs/math/0510447">Non-crossing Partitions under Rotation and Reflection</a>, arXiv:math/0510447 [math.CO], 2005.
%H L. Smiley, <a href="http://www.math.uaa.alaska.edu/~smiley/nc/10NC.pdf">a(5) = 0</a>
%H L. Smiley, <a href="http://www.math.uaa.alaska.edu/~smiley/nc/6pointNCPP.pdf">a(6)=8/2=4</a>
%F a(n) = (A054357(n) - A001405(n))/2.
%t a[n_] := If[n < 6, 0, ((Binomial[2n, n]/(n+1) + DivisorSum[n, Binomial[2#, #] EulerPhi[n/#] Boole[# < n]&])/n - Binomial[n, Floor[n/2]])/2];
%t Array[a, 22] (* _Jean-François Alcover_, Feb 17 2019 *)
%Y Cf. A001405, A054357, A111275.
%K nonn,more
%O 1,6
%A _David Callan_ and _Len Smiley_, Oct 21 2005
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