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%I #27 Apr 05 2019 12:22:31
%S 1,0,1,4,31,288,3272,43580,666143,11491696,220875237,4681264432,
%T 108475235444,2728591657920,74051386322580,2156865088819692,
%U 67113404608820943,2221948578439255200,77990056655776149179
%N Crossing matchings: linear chord diagrams with 2n nodes and n arcs in which each arc crosses another arc.
%H Olivia Beckwith, Victor Luo, Stephen J. Miller, Karen Shen, Nicholas Triantafillou, <a href="https://arxiv.org/abs/1112.3719">Distribution of Eigenvalues of Weighted, Structured Matrix Ensembles</a>, arXiv:1112.3719 [math.PR], 2011-2012.
%H Olivia Beckwith, Victor Luo, Stephen J. Miller, Karen Shen, Nicholas Triantafillou, <a href="https://www.emis.de/journals/INTEGERS/papers/p21/p21.Abstract.html">Distribution of Eigenvalues of Weighted, Structured Matrix Ensembles</a>, Electronic Journal of Combinatorial Number Theory, Volume 15 (2015) #A21.
%H M. Klazar, <a href="http://dx.doi.org/10.1016/S0196-8858(02)00528-6">Non-P-recursiveness of numbers of matchings or linear chord diagrams with many crossings</a>, Advances in Appl. Math., Vol. 30 (2003), pp. 126-136.
%H Alexander Stoimenow, <a href="https://refubium.fu-berlin.de/handle/fub188/1637">On enumeration of chord diagrams and asymptotics of Vassiliev invariants</a>, Dissertation, Mathematik und Informatik, University of Berlin, 1998; see chapter 3.
%F The g.f. (a formal power series) F = 1 + x^2 + 4*x^3 + ... satisfies the differential equation F' = (-x^2*F^3 + F - 1)/(2*x^3*F^2 + 2*x^2*F).
%F a(n) is asymptotic to (2n)!/(e 2^n n!). In other words, the probability that a random matching is a crossing matching is asymptotic to 1/e; see Lemma 3.12 of Stoimenow reference. - _Benoit Cloitre_, Apr 18 2003; corrected by _Dean Hickerson_, Apr 21 2003
%e The 4 crossing matchings on nodes 1, 2, ..., 6 are {13, 25, 46}, {14, 25, 36}, {15, 24, 36} and {14, 26, 35}.
%t a[n_] := a[n]=Module[{x, y, z, i}, y=Sum[a[i]x^i, {i, 0, n-1}]+z*x^n+O[x]^(n+1); Solve[D[y, x]==(-1+y-x^2y^3)/(2x^2y(1+x*y)), z][[1, 1, 2]]]
%Y Cf. A000699, A004300.
%K easy,nonn
%O 0,4
%A _Martin Klazar_, Apr 15 2003
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