A094021 -id:A094021 - cp's OEIS frontend

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

Philippe Flajolet, Apr 20 2000

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.

Row sums of A094021.

Cf. A006013, A000108.

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

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 *)

N=33; x='x+O('x^N); Vec(serreverse(x/((1+x)*(1-sqrt(1-4*x))/(2*x)))) \\ Joerg Arndt, May 25 2016

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

A094040 Triangle read by rows: T(n,k) is the number of noncrossing forests with n vertices and k edges.

Original entry on oeis.org

1, 1, 1, 1, 3, 3, 1, 6, 14, 12, 1, 10, 40, 75, 55, 1, 15, 90, 275, 429, 273, 1, 21, 175, 770, 1911, 2548, 1428, 1, 28, 308, 1820, 6370, 13328, 15504, 7752, 1, 36, 504, 3822, 17640, 51408, 93024, 95931, 43263, 1, 45, 780, 7350, 42840, 162792, 406980, 648945, 600875, 246675

Offset: 1

Emeric Deutsch, May 31 2004

T(n,n-1) yields A001764; T(n,n-2) yields A026004.

			Triangle begins:
  1;
  1,  1;
  1,  3,   3;
  1,  6,  14,  12;
  1, 10,  40,  75,   55;
  1, 15,  90, 275,  429,  273;
  1, 21, 175, 770, 1911, 2548, 1428;
  ...
T(3,1)=3 because the noncrossing forests on 3 vertices A,B,C and having one edge are (A, BC), (B, CA) and (C, AB).

Andrew Howroyd, Table of n, a(n) for n = 1..1275
P. Flajolet and M. Noy, Analytic combinatorics of noncrossing configurations, Discrete Math. 204 (1999), 203-229.

Cf. A001764, A026004, A045739, A094021.

T:=proc(n,k) if k<=n-1 then binomial(n,k+1)*binomial(n+2*k-1,k)/(n+k) else 0 fi end: seq(seq(T(n,k),k=0..n-1),n=1..11);

T[n_, k_] := Binomial[n, k+1] Binomial[n+2k-1, k]/(n+k);
Table[T[n, k], {n, 1, 11}, {k, 0, n-1}] // Flatten (* Jean-François Alcover, Jul 29 2018 *)

T(n,k)=binomial(n, k+1)*binomial(n+2*k-1, k)/(n+k);
for(n=1, 10, for(k=0, n-1, print1(T(n, k), ", ")); print); \\ Andrew Howroyd, Nov 17 2017

T(n, k)=binomial(n, k+1)*binomial(n+2k-1, k)/(n+k) (0<=k<=n-1).

cp's OEIS Frontend

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

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