|
%I #14 Sep 21 2017 03:00:52
%S 1,3,12,51,218,926,3902,16323,67866,280746,1156576,4748398,19439332,
%T 79391708,323584322,1316578403,5348814842,21702312818,87955584152,
%U 356114261498,1440568977932,5822909703908,23520345224732
%N a(n) = 2*4^(n-1) - (3n-1)/(2n+2)*C(2n,n).
%C Cardinality of the set of nesting-similarity classes.
%C Number of Lyngsø-Pedersen structures with n arcs [Saule et al., Theorem 1]. - _Eric M. Schmidt_, Sep 20 2017
%H M. Klazar, <a href="https://arxiv.org/abs/math/0503012">On identities concerning the numbers of crossings and nestings of two edges in matchings</a>, arXiv:math/0503012 [math.CO], 2005.
%H Cédric Saule, Mireille Regnier, Jean-Marc Steyaert, Alain Denise, <a href="https://dmtcs.episciences.org/2834">Counting RNA pseudoknotted structures (extended abstract)</a>, dmtcs:2834 - Discrete Mathematics & Theoretical Computer Science, January 1, 2010, DMTCS Proceedings vol. AN, 22nd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2010)
%F G.f.: C+z^2(2zC'+C)^2C, with C(z) the g.f. of the Catalan numbers.
%F G.f.: (x*(8*x+5*Sqrt[1-4 x]-9)-2*Sqrt[1-4 x]+2)/(2*(1-4*x)*x^2). [_Harvey P. Dale_, Oct 03 2011]
%F Conjecture: 2*(n+1)*a(n) +(-21*n+1)*a(n-1) +2*(36*n-43)*a(n-2) +40*(-2*n+5)*a(n-3)=0. - _R. J. Mathar_, Jun 08 2016
%t Table[2 4^(n-1)-(3n-1)/(2n+2) Binomial[2n,n],{n,30}] (* _Harvey P. Dale_, Oct 03 2011 *)
%Y Equals A006419(n-1) + A000108(n).
%K nonn,easy
%O 1,2
%A _Ralf Stephan_, Apr 17 2005
|
