%I #70 Sep 11 2024 13:11:08

%S 1,1,2,5,17,79,554,5283,65346,966156,16411700,312700297,6589356711,

%T 152041845075,3811786161002,103171594789775,2998419746654530,

%U 93127358763431113,3078376375601255821,107905191542909828013,3997887336845307589431

%N Number of pairings on a bracelet; number of chord diagrams that can be turned over and having n chords.

%C Place 2n points equally spaced on a circle. Draw lines to pair up all the points so that each point has exactly one partner. Allow turning over.

%D R. C. Read, Some Enumeration Problems in Graph Theory. Ph.D. Dissertation, Department of Mathematics, Univ. London, 1958.

%H W. Y.-C. Chen, D. C. Torney, <a href="http://dx.doi.org/10.1016/j.dam.2004.02.013">Equivalence classes of matchings and lattice-square designs</a>, Discr. Appl. Math. 145 (3) (2005) 349-357.

%H Étienne Ghys, <a href="http://arxiv.org/abs/1612.06373">A Singular Mathematical Promenade</a>, arXiv:1612.06373 [math.GT], 2016-2017. See p. 252.

%H A. Khruzin, <a href="http://arXiv.org/abs/math.CO/0008209">Enumeration of chord diagrams</a>, arXiv:math/0008209 [math.CO], 2000.

%H V. A. Liskovets, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL3/LISK/Derseq.html">Some easily derivable sequences</a>, J. Integer Sequences, 3 (2000), #00.2.2.

%H R. J. Mathar, <a href="/A054499/a054499.pdf">Chord Diagrams A054499</a> (2018)

%H R. J. Mathar, <a href="http://vixra.org/abs/1901.0148">Feynman diagrams of the QED vacuum polarization</a>, vixra:1901.0148 (2019)

%H R. C. Read, <a href="/A002831/a002831.pdf">Letter to N. J. A. Sloane, Feb 04 1971</a> (gives initial terms of this sequence)

%H Alexander Stoimenow, <a href="https://doi.org/10.1016/S0012-365X(99)00347-7">On the number of chord diagrams</a>, Discr. Math. 218 (2000), 209-233.

%H <a href="/index/Br#bracelets">Index entries for sequences related to bracelets</a>

%F a(n) = (2*A007769(n) + A047974(n) + A047974(n-1))/4 for n > 0.

%e For n=3, there are 5 bracelets with 3 pairs of beads. They are represented by the words aabbcc, aabcbc, aabccb, abacbc, and abcabc. All of the 6!/(2*2*2) = 90 combinations can be derived from these by some combination of relabeling the pairs, rotation, and reflection. So a(3) = 5. - _Michael B. Porter_, Jul 27 2016

%t max = 19;

%t alpha[p_, q_?EvenQ] := Sum[Binomial[p, 2*k]*q^k*(2*k-1)!!, {k, 0, max}];

%t alpha[p_, q_?OddQ] := q^(p/2)*(p-1)!!;

%t a[0] = 1;

%t a[n_] := 1/4*(Abs[HermiteH[n-1, I/2]] + Abs[HermiteH[n, I/2]] + (2*Sum[Block[{q = (2*n)/p}, alpha[p, q]*EulerPhi[q]], {p, Divisors[ 2*n]}])/(2*n));

%t Table[a[n], {n, 0, max}] (* _Jean-François Alcover_, Sep 05 2013, after _R. J. Mathar_; corrected by _Andrey Zabolotskiy_, Jul 27 2016 *)

%Y Cf. A007769, A104256, A279207, A279208, A003437 (loopless chord diagrams), A322176 (marked chords), A362657, A362658, A362659 (three, four, five instances of each color rather than two), A371305 (Multiset Transf.), A260847 (directed chords).

%K nonn,easy,nice

%O 0,3

%A _Christian G. Bower_, Apr 06 2000 based on a problem by _Wouter Meeussen_

%E Corrected and extended by _N. J. A. Sloane_, Oct 29 2006

%E a(0)=1 prepended back again by _Andrey Zabolotskiy_, Jul 27 2016