login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A007474 Number of circular chord diagrams with n chords, up to rotational symmetry.
(Formerly M1800)
10
1, 0, 1, 2, 7, 36, 300, 3218, 42335, 644808, 11119515, 213865382, 4537496680, 105270612952, 2651295555949, 72042968876506, 2100886276796969, 65446290562491916, 2169090198219290966, 76211647261082309466, 2829612806029873399561 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,4
REFERENCES
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
Dror Bar-Natan, On the Vassiliev Knot Invariants, Topology 34 (1995) 423-472.
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
KEYWORD
nonn,nice
AUTHOR
STATUS
approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified March 28 10:31 EDT 2024. Contains 371240 sequences. (Running on oeis4.)