A007474 Number of circular chord diagrams with n chords, up to rotational symmetry. (Formerly M1800)

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

Offset

0,4

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.

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 *)

Prog

(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

Crossrefs

Sequence in context: A012363 A012717 A072236 * A002724 A348106 A292206

Adjacent sequences: A007471 A007472 A007473 * A007475 A007476 A007477

Keyword

nonn,nice

Author

N. J. A. Sloane

Status

approved