A185100 Dihedral unlabeled Motzkin numbers: number of ways of drawing any number of nonintersecting chords joining n unlabeled points equally spaced on a circle, up to rotations and reflections of the circle.

1, 1, 2, 2, 4, 5, 11, 16, 36, 65, 150, 312, 756, 1743, 4353, 10732, 27489, 70379, 183866, 481952, 1277784, 3402661, 9126689, 24584870, 66567924, 180939737, 493801694, 1352203202, 3715137460, 10237545525, 28291018283, 78384998904, 217715672036, 606103034821, 1691020991782, 4727601528674, 13242641322252, 37162431389051, 104469244613429

Offset: 0

Author

Max Alekseyev, Feb 07 2011

Keywords

• nonn

Comments

Unlabeled version of A001006. Another version is given by A175954.

The number of ways of drawing exactly n chords joining 2n unlabeled points up to rotations and reflections is A006082(n+1). - Andrey Zabolotskiy, May 24 2018

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..200
• Andrew Howroyd, Chord Configuration Symmetries

Crossrefs

Cf. A001006 (labeled points), A175954 (up to rotations only), A175955, A005773, A006082.

Programs

Mathematica

a1006[0] = 1; a1006[n_Integer] := a1006[n] = a1006[n - 1] + Sum[a1006[k]* a1006[n - 2 - k], {k, 0, n - 2}];
a142150[n_] := n*(1 + (-1)^n)/4;
a2426[n_] := Coefficient[(1 + x + x^2)^n, x, n];
a175954[0] = 1; a175954[n_] := (1/n)*(a1006[n] + a142150[n]*a1006[n/2 - 1] + Sum[EulerPhi[n/d]*a2426[d], {d, Most @Divisors[n]}]);
a5773[0] = 1; a5773[n_] := Sum[k/n*Sum[Binomial[n, j]*Binomial[j, 2*j - n - k], {j, 0, n}], {k, 1, n}];
a[0] = 1;
a[n_?OddQ] := With[{m = (n-1)/2}, (1/2)*(a175954[2*m + 1] + a5773[m + 1])];
a[n_?EvenQ] := With[{m = n/2}, (1/4)*(2*a175954[2*m] + a5773[m] + a5773[m + 1] + a1006[m - 1])];
Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Jul 02 2018, after Andrew Howroyd *)

Formula

a(2n+1) = (1/2) * (A175954(2n+1) + A005773(n+1)). - Andrew Howroyd, Apr 01 2017

a(2n) = (1/4) * (2 * A175954(2n) + A005773(n) + A005773(n+1) + A001006(n-1)) for n > 0. - Andrew Howroyd, Apr 01 2017