A185100 - OEIS

%I #20 Jul 02 2018 10:15:38

%S 1,1,2,2,4,5,11,16,36,65,150,312,756,1743,4353,10732,27489,70379,

%T 183866,481952,1277784,3402661,9126689,24584870,66567924,180939737,

%U 493801694,1352203202,3715137460,10237545525,28291018283,78384998904,217715672036,606103034821,1691020991782,4727601528674,13242641322252,37162431389051,104469244613429

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

%C 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 Andrew Howroyd, <a href="/A185100/b185100.txt">Table of n, a(n) for n = 0..200</a>

%H Andrew Howroyd, <a href="/A175954/a175954.txt">Chord Configuration Symmetries</a>

%F a(2n+1) = (1/2) * (A175954(2n+1) + A005773(n+1)). - _Andrew Howroyd_, Apr 01 2017

%F a(2n) = (1/4) * (2 * A175954(2n) + A005773(n) + A005773(n+1) + A001006(n-1)) for n > 0. - _Andrew Howroyd_, Apr 01 2017

%t a1006[0] = 1; a1006[n_Integer] := a1006[n] = a1006[n - 1] + Sum[a1006[k]* a1006[n - 2 - k], {k, 0, n - 2}];

%t a142150[n_] := n*(1 + (-1)^n)/4;

%t a2426[n_] := Coefficient[(1 + x + x^2)^n, x, n];

%t 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 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 a[0] = 1;

%t a[n_?OddQ] := With[{m = (n-1)/2}, (1/2)*(a175954[2*m + 1] + a5773[m + 1])];

%t a[n_?EvenQ] := With[{m = n/2}, (1/4)*(2*a175954[2*m] + a5773[m] + a5773[m + 1] + a1006[m - 1])];

%t Table[a[n], {n, 0, 40}] (* _Jean-François Alcover_, Jul 02 2018, after _Andrew Howroyd_ *)

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

%K nonn

%O 0,3

%A _Max Alekseyev_, Feb 07 2011