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

0,3

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

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, d<n} totient(n/d) * A002426(d)). - Andrew Howroyd, Apr 01 2017

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 *)

Crossrefs

Cf. A001006, A185100, A002426, A002995.

Sequence in context: A032258 A153949 A302400 * A371162 A158649 A357886

Adjacent sequences: A175951 A175952 A175953 * A175955 A175956 A175957

Keyword

nonn

Author

Max Alekseyev, Oct 29 2010

Status

approved