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 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. 4
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 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 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 MATHEMATICA a1006 = 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 = 1; a175954[n_] := (1/n)*(a1006[n] + a142150[n]*a1006[n/2 - 1] + Sum[EulerPhi[n/d]*a2426[d], {d, Most @Divisors[n]}]); a5773 = 1; a5773[n_] := Sum[k/n*Sum[Binomial[n, j]*Binomial[j, 2*j - n - k], {j, 0, n}], {k, 1, n}]; a = 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 *) CROSSREFS Cf. A001006 (labeled points), A175954 (up to rotations only), A175955, A005773, A006082. Sequence in context: A290436 A338048 A127825 * A103420 A329701 A032258 Adjacent sequences:  A185097 A185098 A185099 * A185101 A185102 A185103 KEYWORD nonn AUTHOR Max Alekseyev, Feb 07 2011 STATUS approved

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Last modified May 13 05:02 EDT 2021. Contains 343836 sequences. (Running on oeis4.)