0,3

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

Joe Sawada, A fast algorithm for generating nonisomorphic chord diagrams, SIAM J. Discrete Math, Vol. 15, No. 4, 2002, pp. 546-561.

D. Bar-Natan, On the Vassiliev Knot Invariants, Topology 34 (1995) 423-472.

D. Bar-Natan, Bibliography of Vassiliev Invariants

Combinatorial Object Server, Informationon chord diagrams

A. Khruzin, Enumeration of chord diagrams

Index entries for sequences related to necklaces

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.

(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)) ; ) ; }

Cf. A054499, A104255.

Sequence in context: A007127 A005639 A093730 this_sequence A005805 A058338 A006896

Adjacent sequences: A007766 A007767 A007768 this_sequence A007770 A007771 A007772

nonn,easy,nice

Jean.Betrema(AT)labri.u-bordeaux.fr

More terms from Christian G. Bower (bowerc(AT)usa.net), Apr 06 2000

Corrected and extended by R. J. Mathar, Oct 26 2006