A022494 Number of connected regular linearized chord diagrams of degree n.

1, 1, 1, 2, 5, 16, 63, 293, 1561, 9321, 61436, 442134, 3446077, 28905485, 259585900, 2485120780, 25267283367, 271949606805, 3089330120711, 36943477086287, 463943009361687, 6105064699310785, 84011389289865102

Offset: 0

Author

Alexander Stoimenow (stoimeno(AT)math.toronto.edu)

Keywords

• nonn

Links

• Gheorghe Coserea, Table of n, a(n) for n = 0..202
• A. Stoimenow, Enumeration of chord diagrams and an upper bound for Vassiliev invariants, J. Knot Theory Ramifications, 7 (1998), no. 1, 93-114. [broken link], [DOI]
• Don Zagier, Vassiliev invariants and a strange identity related to the Dedekind eta-function, Topology, vol.40, pp.945-960 (2001); see p.955.

Crossrefs

Cf. A137251.

Programs

PARI

A137251_seq(N) = {
  my(x='x + O('x^(N+1)), t='t+O('t^(N+2)), q=1-x, z=1/t-1, p=vector(N+1));
  p[1]=1; for (n=1, #p-1, p[n+1] = p[n] * (1-q^n)/(1+z*q^n));
  apply(p->Vecrev(p), Vec((apply(p->Pol(p,'t), vecsum(p)/(1+z))-'t)/'t^2));
};
A022494_seq(N) = {
  my(s = 't+'t^2*'x*Ser(apply(v->Polrev(v,'t), A137251_seq(N))),
     r = Ser(vector(N+1, n, subst(polcoeff(s, n-1, 't), 'x, 'u + O('u^(N+1)))),'t));
  Vec(1+subst(Pol(t/serreverse(r) - 1,'t),'t,1));
};
A022494_seq(22) \\ Gheorghe Coserea, Nov 01 2017