A175954 Unlabeled (cyclic) Motzkin numbers: number of ways of drawing any number of nonintersecting chords joining n unlabeled points equally spaced on a circle, up to rotations of the circle.

1, 1, 2, 2, 4, 5, 12, 19, 46, 95, 230, 528, 1320, 3219, 8172, 20714, 53478, 138635, 363486, 957858, 2543476, 6788019, 18218772, 49120019, 133036406, 361736109, 987316658, 2703991820, 7429445752, 20473889133, 56579632732, 156766505691

Offset: 0

Author

Max Alekseyev, Oct 29 2010

Keywords

• nonn

Comments

Unlabeled version of A001006.

The number of such chord configurations on 2n vertices with n chords is given by A002995(n+1).

Links

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

Crossrefs

Cf. A001006, A185100, A002426, A002995.

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];
a[0] = 1; a[n_] := (1/n)*(a1006[n]+a142150[n]*a1006[n/2-1] + Sum[EulerPhi[ n/d]*a2426[d], {d, Most @ Divisors[n]}]);
Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Jul 02 2018, after Andrew Howroyd *)

Formula

For odd prime p, a(p) = (A001006(p) - 1)/p + 1.

a(n) = (1/n) * (A001006(n) + A142150(n) * A001006(n/2-1) + Sum{d|n, dA002426(d)). - Andrew Howroyd, Apr 01 2017