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

1,2

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.

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

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 *)

Prog

(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

Crossrefs

Row sums of A094021.

Cf. A006013, A000108.

Sequence in context: A369300 A055724 A301433 * A086618 A224769 A302285

Adjacent sequences: A054724 A054725 A054726 * A054728 A054729 A054730

Keyword

nonn,easy

Author

Philippe Flajolet, Apr 20 2000

Status

approved