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A007769
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Number of chord diagrams with n chords; number of pairings on a necklace.
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10
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1, 1, 2, 5, 18, 105, 902, 9749, 127072, 1915951, 32743182, 624999093, 13176573910, 304072048265, 7623505722158, 206342800616597, 5996837126024824, 186254702826289089, 6156752656678674792, 215810382466145354405, 7995774669504366055054
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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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.
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REFERENCES
| Joe Sawada, A fast algorithm for generating nonisomorphic chord diagrams, SIAM J. Discrete Math, Vol. 15, No. 4, 2002, pp. 546-561.
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LINKS
| D. Bar-Natan, On the Vassiliev Knot Invariants, Topology 34 (1995) 423-472.
D. Bar-Natan, Bibliography of Vassiliev Invariants.
W. Y.-C. Chen, D. C. Torney, Equivalence classes of matchings and lattice-square designs, Discr. Appl. Math. 145 (3) (2005) 349-357., table of C_2n.
Combinatorial Object Server, Informationon chord diagrams
A. Khruzin, Enumeration of chord diagrams arXiv:math.CO/0008209
Index entries for sequences related to necklaces
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FORMULA
| 2n a_n = sum_{2n=pq} alpha(p, q)phi(q), phi = Euler function, alpha(p, q) = sum_{k >= 0} binomial(p, 2k) q^k (2k-1)!! if q even, = q^{p/2} (p-1)!! if q odd.
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PROG
| (PARI program from R. J. Mathar) doublefactorial(n)={ local(resul) ; resul=1 ; forstep(i=n, 2, -2, resul *= i ; ) ; return(resul) ; }
alpha(n, q)={ if(q %2, return( q^(p/2)*doublefactorial(p-1)), return( sum(k=0, p/2, binomial(p, 2*k)*q^k*doublefactorial(2*k-1)) ) ; ) ; }
A007769(n)={ local(resul, q) ; if(n==0, return(1), resul=0 ; fordiv(2*n, p, q=2*n/p ; resul += alpha(p, q)*eulerphi(q) ; ); return(resul/(2*n)) ; ) ; } { for(n=0, 20, print(n, " ", A007769(n)) ; ) ; }
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CROSSREFS
| Cf. A054499, A104255.
Sequence in context: A007127 A005639 A093730 * A174122 A005805 A058338
Adjacent sequences: A007766 A007767 A007768 * A007770 A007771 A007772
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KEYWORD
| nonn,easy,nice
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AUTHOR
| Jean.Betrema(AT)labri.u-bordeaux.fr
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EXTENSIONS
| More terms from Christian G. Bower (bowerc(AT)usa.net), Apr 06 2000
Corrected and extended by R. J. Mathar, Oct 26 2006
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