A054727 Number of forests of rooted trees with n nodes on a circle without crossing edges.
1, 2, 7, 33, 181, 1083, 6854, 45111, 305629, 2117283, 14929212, 106790500, 773035602, 5652275723, 41683912721, 309691336359, 2315772552485, 17415395593371, 131632335068744, 999423449413828, 7618960581522348, 58295017292748756, 447517868947619432, 3445923223190363608
Offset: 1
Links
- Andrew Howroyd, Table of n, a(n) for n = 1..200
- C. Banderier and D. Merlini, Lattice paths with an infinite set of jumps, FPSAC02, Melbourne, 2002.
- F. Cazals, Combinatorics of Non-Crossing Configurations, Studies in Automatic Combinatorics, Volume II (1997).
- Guillermo Esteban, Clemens Huemer, Rodrigo I. Silveira, New production matrices for geometric graphs, arXiv:2003.00524 [math.CO], 2020.
- Philippe Flajolet, Source
- Philippe Flajolet, Enumeration of planar configurations in computational geometry
- P. Flajolet and M. Noy, Analytic combinatorics of noncrossing configurations, Discrete Math. 204 (1999), 203-229.
- P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 486, 502
- Samuele Giraudo, Combalgebraic structures on decorated cliques, Formal Power Series and Algebraic Combinatorics, Séminaire Lotharingien de Combinatoire, 78B.15, 2017, p. 8, arXiv:1709.08416 [math.CO], 2017.
- Anssi Yli-Jyrä and Carlos Gómez-Rodríguez, Generic Axiomatization of Families of Noncrossing Graphs in Dependency Parsing, arXiv:1706.03357 [cs.CL], 2017.
Programs
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Maple
ZZ:=[F,{F=Union(Epsilon,ZB),ZB=Prod(Z1,P),P=Sequence(B),B=Prod(P,Z1,P),Z1=Prod(Z,F)}, unlabeled]: seq(count(ZZ,size=n),n=1..20); # Zerinvary Lajos, Apr 22 2007 -
Mathematica
a[n_] := (3*n-3)!/((n-1)!*(2*n-1)!)*HypergeometricPFQ[{1-2*n, 1-n, -n}, {3/2 - 3*n/2, 2 - 3*n/2}, -1/4]; Table[a[n], {n, 1, 20}] (* Jean-François Alcover, Sep 05 2012, after formula *) -
PARI
N=33; x='x+O('x^N); Vec(serreverse(x/((1+x)*(1-sqrt(1-4*x))/(2*x)))) \\ Joerg Arndt, May 25 2016
Formula
a(n) = Sum_{j=1..n} binomial(n, j-1) * binomial(3*n-2*j-1, n-j) / (2*n - j).
G.f. A(x) satisfies 2*A(x)^2=x*(1-sqrt(1-4*A(x)))*(1-A(x)). - Vladimir Kruchinin, Nov 25 2011
From Peter Bala, Nov 07 2015: (Start)
O.g.f. A(x) = revert(x/((1 + x)*C(x))), where C(x) = (1 - sqrt(1 - 4*x))/(2*x) is the o.g.f for the Catalan numbers A000108.
Row sums of A094021. (End)
Conjecture: -2(37*n-80) *(n-1) *(2*n-1) *a(n) +2*(592*n^3-3056*n^2+5045*n-2665) *a(n-1) +2*(148*n^3-986*n^2+2021*n-1255) *a(n-2) -5*(n-5) *(n-2) *(37*n-43) *a(n-3)=0. - R. J. Mathar, Apr 30 2018
a(n) ~ c * d^n / (sqrt(Pi) * n^(3/2)), where d = 8.2246915409778560686084627753... is the real root of the equation 5 - 8*d - 32*d^2 + 4*d^3 = 0 and c = 0.07465927842190452347018812862935237... is the positive real root of the equation -125 + 22376*c^2 + 8880*c^4 + 592*c^6 = 0. - Vaclav Kotesovec, Apr 30 2018