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 A081054 Crossing matchings: linear chord diagrams with 2n nodes and n arcs in which each arc crosses another arc. 1
 1, 0, 1, 4, 31, 288, 3272, 43580, 666143, 11491696, 220875237, 4681264432, 108475235444, 2728591657920, 74051386322580, 2156865088819692, 67113404608820943, 2221948578439255200, 77990056655776149179 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 LINKS Table of n, a(n) for n=0..18. Olivia Beckwith, Victor Luo, Stephen J. Miller, Karen Shen, Nicholas Triantafillou, Distribution of Eigenvalues of Weighted, Structured Matrix Ensembles, arXiv:1112.3719 [math.PR], 2011-2012. Olivia Beckwith, Victor Luo, Stephen J. Miller, Karen Shen, Nicholas Triantafillou, Distribution of Eigenvalues of Weighted, Structured Matrix Ensembles, Electronic Journal of Combinatorial Number Theory, Volume 15 (2015) #A21. M. Klazar, Non-P-recursiveness of numbers of matchings or linear chord diagrams with many crossings, Advances in Appl. Math., Vol. 30 (2003), pp. 126-136. Alexander Stoimenow, On enumeration of chord diagrams and asymptotics of Vassiliev invariants, Dissertation, Mathematik und Informatik, University of Berlin, 1998; see chapter 3. FORMULA 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). 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 EXAMPLE The 4 crossing matchings on nodes 1, 2, ..., 6 are {13, 25, 46}, {14, 25, 36}, {15, 24, 36} and {14, 26, 35}. MATHEMATICA 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]]] CROSSREFS Cf. A000699, A004300. Sequence in context: A077615 A039306 A265949 * A261053 A192407 A000858 Adjacent sequences: A081051 A081052 A081053 * A081055 A081056 A081057 KEYWORD easy,nonn AUTHOR Martin Klazar, Apr 15 2003 STATUS approved

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Last modified December 9 05:43 EST 2023. Contains 367685 sequences. (Running on oeis4.)