cp's OEIS Frontend

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

%I A022494 #19 Nov 03 2017 03:45:08
%S A022494 1,1,1,2,5,16,63,293,1561,9321,61436,442134,3446077,28905485,
%T A022494 259585900,2485120780,25267283367,271949606805,3089330120711,
%U A022494 36943477086287,463943009361687,6105064699310785,84011389289865102
%N A022494 Number of connected regular linearized chord diagrams of degree n.
%H A022494 Gheorghe Coserea, <a href="/A022494/b022494.txt">Table of n, a(n) for n = 0..202</a>
%H A022494 A. Stoimenow, <a href="http://www.math.toronto.edu/stoimeno/bound.ps.gz">Enumeration of chord diagrams and an upper bound for Vassiliev invariants</a>, J. Knot Theory Ramifications, 7 (1998), no. 1, 93-114. [broken link], <a href="http://dx.doi.org/10.1142/S0218216598000073">[DOI]</a>
%H A022494 Don Zagier, <a href="http://people.mpim-bonn.mpg.de/zagier/">Vassiliev invariants and a strange identity related to the Dedekind eta-function</a>, Topology, vol.40, pp.945-960 (2001); see p.955.
%o A022494 (PARI)
%o A022494 A137251_seq(N) = {
%o A022494   my(x='x + O('x^(N+1)), t='t+O('t^(N+2)), q=1-x, z=1/t-1, p=vector(N+1));
%o A022494   p[1]=1; for (n=1, #p-1, p[n+1] = p[n] * (1-q^n)/(1+z*q^n));
%o A022494   apply(p->Vecrev(p), Vec((apply(p->Pol(p,'t), vecsum(p)/(1+z))-'t)/'t^2));
%o A022494 };
%o A022494 A022494_seq(N) = {
%o A022494   my(s = 't+'t^2*'x*Ser(apply(v->Polrev(v,'t), A137251_seq(N))),
%o A022494      r = Ser(vector(N+1, n, subst(polcoeff(s, n-1, 't), 'x, 'u + O('u^(N+1)))),'t));
%o A022494   Vec(1+subst(Pol(t/serreverse(r) - 1,'t),'t,1));
%o A022494 };
%o A022494 A022494_seq(22) \\ _Gheorghe Coserea_, Nov 01 2017
%Y A022494 Cf. A137251.
%K A022494 nonn
%O A022494 0,4
%A A022494 Alexander Stoimenow (stoimeno(AT)math.toronto.edu)