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.
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..200
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, Information on Chord Diagrams
Étienne Ghys, A Singular Mathematical Promenade, arXiv:1612.06373 [math.GT], 2016. See p. 252.
A. Khruzin, Enumeration of chord diagrams, arXiv:math/0008209 [math.CO], 2000.
R. J. Mathar, Feynman diagrams of the QED vacuum polarization, vixra:1901.0148 (2019), Section V.
Joe Sawada, A fast algorithm for generating nonisomorphic chord diagrams, SIAM J. Discrete Math, Vol. 15, No. 4, 2002, pp. 546-561.
Alexander Stoimenow, On the number of chord diagrams, Discr. Math. 218 (2000), 209-233.
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.
MAPLE
with(numtheory):
alpha:=proc(p, q):if is(q, even) then
add(binomial(p, 2*k)*q^k*doublefactorial(2*k-1), k=0..p/2)
else q^(p/2)*doublefactorial(p-1) fi end:
a:=n->add(alpha(2*n/p, p)*phi(p), p=divisors(2*n))/2/n:
a(0):=1:seq(a(k), k=0..20); # Robert FERREOL, Oct 10 2018
MATHEMATICA
max = 20; alpha[p_, q_?EvenQ] := Sum[Binomial[p, 2k]*q^k*(2k-1)!!, {k, 0, max}]; alpha[p_, q_?OddQ] := q^(p/2)*(p-1)!!; a[0] = 1; a[n_] := Sum[q = 2n/p; alpha[p, q]*EulerPhi[q], {p, Divisors[2n]}]/(2n); Table[a[n], {n, 0, max}] (* Jean-François Alcover, May 07 2012, after R. J. Mathar *)
Stoimenow states that a Mma package is available from his website. - N. J. A. Sloane, Jul 26 2018
PROG
(PARI) 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)) ; ) ; } \\ R. J. Mathar, Oct 26 2006
CROSSREFS
KEYWORD
nonn,easy,nice
AUTHOR
Jean.Betrema(AT)labri.u-bordeaux.fr
EXTENSIONS
More terms from Christian G. Bower, Apr 06 2000
Corrected and extended by R. J. Mathar, Oct 26 2006
STATUS
approved