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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.

%I A185100 #20 Jul 02 2018 10:15:38
%S A185100 1,1,2,2,4,5,11,16,36,65,150,312,756,1743,4353,10732,27489,70379,
%T A185100 183866,481952,1277784,3402661,9126689,24584870,66567924,180939737,
%U A185100 493801694,1352203202,3715137460,10237545525,28291018283,78384998904,217715672036,606103034821,1691020991782,4727601528674,13242641322252,37162431389051,104469244613429
%N 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.
%C A185100 Unlabeled version of A001006. Another version is given by A175954.
%C A185100 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
%H A185100 Andrew Howroyd, <a href="/A185100/b185100.txt">Table of n, a(n) for n = 0..200</a>
%H A185100 Andrew Howroyd, <a href="/A175954/a175954.txt">Chord Configuration Symmetries</a>
%F A185100 a(2n+1) = (1/2) * (A175954(2n+1) + A005773(n+1)). - _Andrew Howroyd_, Apr 01 2017
%F A185100 a(2n) = (1/4) * (2 * A175954(2n) + A005773(n) + A005773(n+1) + A001006(n-1)) for n > 0. - _Andrew Howroyd_, Apr 01 2017
%t A185100 a1006[0] = 1; a1006[n_Integer] := a1006[n] = a1006[n - 1] + Sum[a1006[k]* a1006[n - 2 - k], {k, 0, n - 2}];
%t A185100 a142150[n_] := n*(1 + (-1)^n)/4;
%t A185100 a2426[n_] := Coefficient[(1 + x + x^2)^n, x, n];
%t A185100 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]}]);
%t A185100 a5773[0] = 1; a5773[n_] := Sum[k/n*Sum[Binomial[n, j]*Binomial[j, 2*j - n - k], {j, 0, n}], {k, 1, n}];
%t A185100 a[0] = 1;
%t A185100 a[n_?OddQ] := With[{m = (n-1)/2}, (1/2)*(a175954[2*m + 1] + a5773[m + 1])];
%t A185100 a[n_?EvenQ] := With[{m = n/2}, (1/4)*(2*a175954[2*m] + a5773[m] + a5773[m + 1] + a1006[m - 1])];
%t A185100 Table[a[n], {n, 0, 40}] (* _Jean-François Alcover_, Jul 02 2018, after _Andrew Howroyd_ *)
%Y A185100 Cf. A001006 (labeled points), A175954 (up to rotations only), A175955, A005773, A006082.
%K A185100 nonn
%O A185100 0,3
%A A185100 _Max Alekseyev_, Feb 07 2011