

A054499


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


20



1, 1, 2, 5, 17, 79, 554, 5283, 65346, 966156, 16411700, 312700297, 6589356711, 152041845075, 3811786161002, 103171594789775, 2998419746654530, 93127358763431113, 3078376375601255821, 107905191542909828013, 3997887336845307589431
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OFFSET

0,3


COMMENTS

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.


REFERENCES

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


LINKS



FORMULA



EXAMPLE

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


MATHEMATICA

max = 19;
alpha[p_, q_?EvenQ] := Sum[Binomial[p, 2*k]*q^k*(2*k1)!!, {k, 0, max}];
alpha[p_, q_?OddQ] := q^(p/2)*(p1)!!;
a[0] = 1;
a[n_] := 1/4*(Abs[HermiteH[n1, 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));


CROSSREFS

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.).


KEYWORD

nonn,easy,nice


AUTHOR



EXTENSIONS



STATUS

approved



