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A054727 Number of forests of rooted trees with n nodes on a circle without crossing edges. 7

%I

%S 1,2,7,33,181,1083,6854,45111,305629,2117283,14929212,106790500,

%T 773035602,5652275723,41683912721,309691336359,2315772552485,

%U 17415395593371,131632335068744,999423449413828,7618960581522348,58295017292748756,447517868947619432,3445923223190363608

%N Number of forests of rooted trees with n nodes on a circle without crossing edges.

%H Andrew Howroyd, <a href="/A054727/b054727.txt">Table of n, a(n) for n = 1..200</a>

%H C. Banderier and D. Merlini, <a href="http://algo.inria.fr/banderier/Papers/infjumps.ps">Lattice paths with an infinite set of jumps</a>, FPSAC02, Melbourne, 2002.

%H F. Cazals, <a href="http://algo.inria.fr/libraries/autocomb/NCC-html/NCC.html">Combinatorics of Non-Crossing Configurations</a>, Studies in Automatic Combinatorics, Volume II (1997).

%H Guillermo Esteban, Clemens Huemer, Rodrigo I. Silveira, <a href="https://arxiv.org/abs/2003.00524">New production matrices for geometric graphs</a>, arXiv:2003.00524 [math.CO], 2020.

%H Philippe Flajolet, <a href="http://algo.inria.fr/flajolet/Publications">Source</a>

%H Philippe Flajolet, <a href="http://algo.inria.fr/libraries/autocomb/nc-configs-html/nc-configs1.html">Enumeration of planar configurations in computational geometry</a>

%H P. Flajolet and M. Noy, <a href="https://dx.doi.org/10.1016/S0012-365X(98)00372-0">Analytic combinatorics of noncrossing configurations</a>, Discrete Math. 204 (1999), 203-229.

%H P. Flajolet and R. Sedgewick, <a href="http://algo.inria.fr/flajolet/Publications/books.html">Analytic Combinatorics</a>, 2009; see page 486, 502

%H Samuele Giraudo, <a href="https://arxiv.org/abs/1709.08416">Combalgebraic structures on decorated cliques</a>, Formal Power Series and Algebraic Combinatorics, Séminaire Lotharingien de Combinatoire, 78B.15, 2017, p. 8, arXiv:1709.08416 [math.CO], 2017.

%H Anssi Yli-Jyrä and Carlos Gómez-Rodríguez, <a href="https://arxiv.org/abs/1706.03357">Generic Axiomatization of Families of Noncrossing Graphs in Dependency Parsing</a>, arXiv:1706.03357 [cs.CL], 2017.

%F a(n) = Sum_{j=1..n} binomial(n, j-1) * binomial(3*n-2*j-1, n-j) / (2*n - j).

%F G.f. A(x) satisfies 2*A(x)^2=x*(1-sqrt(1-4*A(x)))*(1-A(x)). - _Vladimir Kruchinin_, Nov 25 2011

%F From _Peter Bala_, Nov 07 2015: (Start)

%F 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.

%F Row sums of A094021. (End)

%F 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

%F 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

%p 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

%t 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 *)

%o (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

%Y Row sums of A094021.

%Y Cf. A006013, A000108.

%K nonn,easy

%O 1,2

%A _Philippe Flajolet_, Apr 20 2000

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Last modified March 8 19:37 EST 2021. Contains 341953 sequences. (Running on oeis4.)