A007474 Number of circular chord diagrams with n chords, up to rotational symmetry.

1, 0, 1, 2, 7, 36, 300, 3218, 42335, 644808, 11119515, 213865382, 4537496680, 105270612952, 2651295555949, 72042968876506, 2100886276796969, 65446290562491916, 2169090198219290966, 76211647261082309466, 2829612806029873399561

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• nice

References

• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Gheorghe Coserea, Table of n, a(n) for n = 0..300
• Dror Bar-Natan, On the Vassiliev Knot Invariants, Topology 34 (1995) 423-472.
• D. Bar-Natan, Bibliography of Vassiliev Invariants
• E. Krasko, A. Omelchenko, Enumeration of Chord Diagrams without Loops and Parallel Chords, arXiv preprint arXiv:1601.05073 [math.CO], 2016.
• E. Krasko, A. Omelchenko, Enumeration of Chord Diagrams without Loops and Parallel Chords, The Electronic Journal of Combinatorics, 24(3) (2017), #P3.43.

Programs

Mathematica

m = 20; Clear[M]; M[, ] = 0; Mget[n_, k_] := Which[n < 0, 0, n == 0, 1, n == 1, 1 - Mod[k, 2], n == 2, k - Mod[k, 2], True, M[n, k]]; Mset[n_, k_, v_] := (M[n, k] = v); Minit[] = (tmp = 0; For[n = 3, n <= 2*m, n++, For[k = 1, k <= 2*m, k++, tmp = If[Mod[k, 2] == 1, k*(n-1)*Mget[n-2, k] + Mget[n-4, k], Mget[n-1, k] + k*(n-1) * Mget[n-2, k] - Mget[n-3, k] + Mget[n-4, k]]; Mset[n, k, tmp]]];); a[n_] := DivisorSum[2*n, EulerPhi[#] * (Mget[2*n/#, #] - Mget[2*n/# - 2, #])&] / (2*n); Minit[]; Prepend[ Array[a, m], 1] (* Jean-François Alcover, Apr 24 2017, after Gheorghe Coserea *)

PARI

N = 20; M = matrix(2*N, 2*N);
Mget(n,k) = { if (n<0, 0, n==0, 1, n==1, 1-(k%2), n==2, k-(k%2), M[n,k]) };
Mset(n,k,v) = { M[n,k] = v;};
Minit() = {
  my(tmp = 0);
  for (n=3, 2*N, for(k=1, 2*N,
    tmp = if (k%2, k*(n-1) * Mget(n-2, k) + Mget(n-4, k),
    Mget(n-1, k) + k*(n-1) * Mget(n-2, k) - Mget(n-3, k) + Mget(n-4, k));
    Mset(n, k, tmp)));
};
a(n) = sumdiv(2*n, d, eulerphi(d) * (Mget(2*n/d, d) - Mget(2*n/d-2, d))) / (2*n);
Minit();
concat(1, vector(N, n, a(n)))  \\ Gheorghe Coserea, Dec 10 2016