cp's OEIS Frontend

A111276 Number of chiral non-crossing partition patterns of n points on a circle, divided by 2.

%I A111276 #15 Nov 19 2024 23:46:46
%S A111276 0,0,0,0,0,4,14,60,210,728,2442,8252,27716,93924,319964,1098900,
%T A111276 3800928,13244836,46460738,164015272,582353976,2078812492,7457141650,
%U A111276 26871707908,97236327900,353213328024,1287648322950,4709765510884,17279999438748,63583033400968
%N A111276 Number of chiral non-crossing partition patterns of n points on a circle, divided by 2.
%C A111276 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 A111276 Andrew Howroyd, <a href="/A111276/b111276.txt">Table of n, a(n) for n = 1..1000</a>
%H A111276 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 A111276 L. Smiley, <a href="http://www.math.uaa.alaska.edu/~smiley/nc/10NC.pdf">a(5) = 0</a>
%H A111276 L. Smiley, <a href="http://www.math.uaa.alaska.edu/~smiley/nc/6pointNCPP.pdf">a(6)=8/2=4</a>
%F A111276 a(n) = (A054357(n) - A001405(n))/2.
%t A111276 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 A111276 Array[a, 22] (* _Jean-François Alcover_, Feb 17 2019 *)
%o A111276 (PARI) a(n) = (sumdiv(n, d, eulerphi(n/d)*binomial(2*d, d))/n - binomial(2*n, n)/(n+1) - binomial(n,n\2))/2 \\ _Andrew Howroyd_, Nov 19 2024
%Y A111276 Cf. A001405, A054357, A111275.
%K A111276 nonn
%O A111276 1,6
%A A111276 _David Callan_ and _Len Smiley_, Oct 21 2005
%E A111276 a(23) onwards from _Andrew Howroyd_, Nov 19 2024