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

0, 0, 0, 0, 0, 4, 14, 60, 210, 728, 2442, 8252, 27716, 93924, 319964, 1098900, 3800928, 13244836, 46460738, 164015272, 582353976, 2078812492, 7457141650, 26871707908, 97236327900, 353213328024, 1287648322950, 4709765510884, 17279999438748, 63583033400968

Offset

1,6

Comments

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.

Links

Andrew Howroyd, Table of n, a(n) for n = 1..1000

D. Callan and L. Smiley, Non-crossing Partitions under Rotation and Reflection, arXiv:math/0510447 [math.CO], 2005.

L. Smiley, a(5) = 0

L. Smiley, a(6)=8/2=4

Formula

a(n) = (A054357(n) - A001405(n))/2.

Mathematica

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];
Array[a, 22] (* Jean-François Alcover, Feb 17 2019 *)

Prog

(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

Crossrefs

Cf. A001405, A054357, A111275.

Sequence in context: A307488 A351634 A241706 * A377148 A149493 A299926

Adjacent sequences: A111273 A111274 A111275 * A111277 A111278 A111279

Keyword

nonn

Author

David Callan and Len Smiley, Oct 21 2005

Extensions

a(23) onwards from Andrew Howroyd, Nov 19 2024

Status

approved