cp's OEIS Frontend

A054499 Number of pairings on a bracelet; number of chord diagrams that can be turned over and having n chords.

%I A054499 #70 Sep 11 2024 13:11:08
%S A054499 1,1,2,5,17,79,554,5283,65346,966156,16411700,312700297,6589356711,
%T A054499 152041845075,3811786161002,103171594789775,2998419746654530,
%U A054499 93127358763431113,3078376375601255821,107905191542909828013,3997887336845307589431
%N A054499 Number of pairings on a bracelet; number of chord diagrams that can be turned over and having n chords.
%C A054499 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 A054499 R. C. Read, Some Enumeration Problems in Graph Theory. Ph.D. Dissertation, Department of Mathematics, Univ. London, 1958.
%H A054499 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 A054499 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 A054499 A. Khruzin, <a href="http://arXiv.org/abs/math.CO/0008209">Enumeration of chord diagrams</a>, arXiv:math/0008209 [math.CO], 2000.
%H A054499 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 A054499 R. J. Mathar, <a href="/A054499/a054499.pdf">Chord Diagrams A054499</a> (2018)
%H A054499 R. J. Mathar, <a href="http://vixra.org/abs/1901.0148">Feynman diagrams of the QED vacuum polarization</a>, vixra:1901.0148 (2019)
%H A054499 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 A054499 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 A054499 <a href="/index/Br#bracelets">Index entries for sequences related to bracelets</a>
%F A054499 a(n) = (2*A007769(n) + A047974(n) + A047974(n-1))/4 for n > 0.
%e A054499 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 A054499 max = 19;
%t A054499 alpha[p_, q_?EvenQ] := Sum[Binomial[p, 2*k]*q^k*(2*k-1)!!, {k, 0, max}];
%t A054499 alpha[p_, q_?OddQ] := q^(p/2)*(p-1)!!;
%t A054499 a[0] = 1;
%t A054499 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 A054499 Table[a[n], {n, 0, max}] (* _Jean-François Alcover_, Sep 05 2013, after _R. J. Mathar_; corrected by _Andrey Zabolotskiy_, Jul 27 2016 *)
%Y A054499 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 A054499 nonn,easy,nice
%O A054499 0,3
%A A054499 _Christian G. Bower_, Apr 06 2000 based on a problem by _Wouter Meeussen_
%E A054499 Corrected and extended by _N. J. A. Sloane_, Oct 29 2006
%E A054499 a(0)=1 prepended back again by _Andrey Zabolotskiy_, Jul 27 2016