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

%I A007474 M1800 #40 Dec 17 2021 11:14:16
%S A007474 1,0,1,2,7,36,300,3218,42335,644808,11119515,213865382,4537496680,
%T A007474 105270612952,2651295555949,72042968876506,2100886276796969,
%U A007474 65446290562491916,2169090198219290966,76211647261082309466,2829612806029873399561
%N A007474 Number of circular chord diagrams with n chords, up to rotational symmetry.
%D A007474 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A007474 Gheorghe Coserea, <a href="/A007474/b007474.txt">Table of n, a(n) for n = 0..300</a>
%H A007474 Dror Bar-Natan, <a href="https://www.math.toronto.edu/drorbn/papers/OnVassiliev/">On the Vassiliev Knot Invariants</a>, Topology 34 (1995) 423-472.
%H A007474 D. Bar-Natan, <a href="http://www.pdmi.ras.ru/~duzhin/VasBib/">Bibliography of Vassiliev Invariants</a>
%H A007474 E. Krasko, A. Omelchenko, <a href="http://arxiv.org/abs/1601.05073">Enumeration of Chord Diagrams without Loops and Parallel Chords</a>, arXiv preprint arXiv:1601.05073 [math.CO], 2016.
%H A007474 E. Krasko, A. Omelchenko, <a href="http://www.combinatorics.org/ojs/index.php/eljc/article/view/v24i3p43">Enumeration of Chord Diagrams without Loops and Parallel Chords</a>, The Electronic Journal of Combinatorics, 24(3) (2017), #P3.43.
%t A007474 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_ *)
%o A007474 (PARI)
%o A007474 N = 20; M = matrix(2*N, 2*N);
%o A007474 Mget(n,k) = { if (n<0, 0, n==0, 1, n==1, 1-(k%2), n==2, k-(k%2), M[n,k]) };
%o A007474 Mset(n,k,v) = { M[n,k] = v;};
%o A007474 Minit() = {
%o A007474   my(tmp = 0);
%o A007474   for (n=3, 2*N, for(k=1, 2*N,
%o A007474     tmp = if (k%2, k*(n-1) * Mget(n-2, k) + Mget(n-4, k),
%o A007474     Mget(n-1, k) + k*(n-1) * Mget(n-2, k) - Mget(n-3, k) + Mget(n-4, k));
%o A007474     Mset(n, k, tmp)));
%o A007474 };
%o A007474 a(n) = sumdiv(2*n, d, eulerphi(d) * (Mget(2*n/d, d) - Mget(2*n/d-2, d))) / (2*n);
%o A007474 Minit();
%o A007474 concat(1, vector(N, n, a(n)))  \\ _Gheorghe Coserea_, Dec 10 2016
%K A007474 nonn,nice
%O A007474 0,4
%A A007474 _N. J. A. Sloane_