A102593 -id:A102593 - cp's OEIS frontend

A102893 Number of noncrossing trees with n edges and having degree of the root at least 2.

1, 0, 1, 5, 25, 130, 700, 3876, 21945, 126500, 740025, 4382625, 26225628, 158331880, 963250600, 5899491640, 36345082425, 225082957512, 1400431689475, 8749779798375, 54874635255825, 345329274848250, 2179969531405680

Offset: 0

Emeric Deutsch, Jan 16 2005

[a(n+2)]= [1,5,25,130,700,...] is the self-convolution 5th power of A001764. - Philippe Deléham, Nov 11 2009

a(n) is the number of dissections of a convex (2n+2)-sided polygon by nonintersecting diagonals into quadrilaterals such that at least one of the dividing diagonals passes through a chosen vertex. - Muhammed Sefa Saydam, Jan 24 2025

			a(2)=1 because among the noncrossing trees with 2 edges, namely /_, _\ and /\, only the last one has root degree >1.

Andrew Howroyd, Table of n, a(n) for n = 0..200
David Bevan, Robert Brignall, Andrew Elvey Price and Jay Pantone, A structural characterisation of Av(1324) and new bounds on its growth rate, arXiv preprint arXiv:1711.10325 [math.CO], 2017-2019.
Emanuele Munarini, Shifting Property for Riordan, Sheffer and Connection Constants Matrices, Journal of Integer Sequences, Vol. 20 (2017), Article 17.8.2.
M. Noy, Enumeration of noncrossing trees on a circle, Discrete Math., 180, 301-313, 1998.

Column k=0 of A102892 and column k=0 of A102593.

Cf. A001764, A006013.

a:=proc(n) if n=0 then 1 else 5*binomial(3*n-1,n-2)/(3*n-1) fi end:
seq(a(n), n=0..25);
# Recurrence:
a := proc(n) option remember; if n < 3 then return [1,0,1][n+1] fi;
(27*n^3 - 81*n^2 + 78*n - 24)*a(n - 1)/(4*n^3 - 6*n^2 - 4*n) end:
seq(a(n), n=0..23); # Peter Luschny, Aug 08 2020
alias(PS=ListTools:-PartialSums): A102893List := proc(m) local A, P, n;
A := [1,0]; P := [1]; for n from 1 to m - 2 do P := PS(PS([op(P), P[-1]]));
A := [op(A), P[-2]] od; A end: A102893List(23); # Peter Luschny, Mar 26 2022

a[0] = 1; a[n_] := 5*Binomial[3n-1, n-2]/(3n-1); Table[a[n], {n, 0, 22}] (* Jean-François Alcover, Mar 01 2018 *)

a(n) = if(n<=1, n==0, 5*binomial(3*n-1, n-2)/(3*n-1)); \\ Andrew Howroyd, Nov 17 2017

a(0)=1; a(n) = 5*binomial(3n-1, n-2)/(3n-1) if n > 0.
G.f.: g - z*g^2, where g = 1 + z*g^3 is the g.f. of the ternary numbers (A001764).
a(n) = A001764(n) - A006013(n-1).
D-finite with recurrence 2*n*(2*n+1)*(n-2)*a(n) -3*(n-1)*(3*n-4)*(3*n-2)*a(n-1)=0. - R. J. Mathar, Feb 16 2018
a(n) ~ (5*3^(3*n + 1/2))/(36*4^n*n^(3/2)*sqrt(Pi)). - Peter Luschny, Aug 08 2020

A102594 Number of noncrossing trees with n edges in which no border edges emanate from the root.

Original entry on oeis.org

1, 0, 0, 1, 7, 42, 245, 1428, 8379, 49588, 296010, 1781325, 10798788, 65900296, 404565252, 2496994136, 15486165555, 96464124648, 603262881620, 3786268349115, 23842082904255, 150586208376450, 953736669989985

Offset: 0

Emeric Deutsch, Jan 22 2005

nonn

			a(2)=0 because in all the three noncrossing trees with 2 edges, namely, /_, /\ and _\, the root (=the top vertex) is incident with at least one border edge.

Andrew Howroyd, Table of n, a(n) for n = 0..200
P. Flajolet and M. Noy, Analytic combinatorics of non-crossing configurations, Discrete Math., 204, 203-229, 1999.
Emanuele Munarini, Shifting Property for Riordan, Sheffer and Connection Constants Matrices, Journal of Integer Sequences, Vol. 20 (2017), Article 17.8.2.
M. Noy, Enumeration of noncrossing trees on a circle, Discrete Math., 180, 301-313, 1998.

Column k=0 of A102593.

Cf. A001764.

a:=n->7/3*(n-1)*(n-2)*binomial(3*n,n)/(3*n-1)/(2*n+1)/(3*n-2): 1,seq(a(n), n=1..25);

a[0] = a[3] = 1; a[n_] := 7*Binomial[3n-3, 2n+1]/(n-3); Table[a[n], {n, 0, 22}] (* Jean-François Alcover, Jan 21 2013 *)

a(n) = if (n==0, 1, 7/3*(n-1)*(n-2)*binomial(3*n, n)/(3*n-1)/(2*n+1)/(3*n-2)); \\ Michel Marcus, Oct 26 2015

Vec((g->g*(1+x-2*x*g))(1+serreverse(x/(1+x)^3 + O(x^30)))) \\ Andrew Howroyd, Nov 17 2017

a(n) = 7/3*(n-1)*(n-2)*binomial(3*n, n)/(3*n-1)/(2*n+1)/(3*n-2) for n > 0; a(0)=1.
G.f.: g*(1+z-2*z*g), where g = 1+z*g^3 is the g.f. of the ternary numbers (A001764).
From Karol A. Penson, Mar 12 2018: (Start)
a(n+3) = 7*binomial(3*n+6, 2*n+6)/(2*n+7).
a(n+3) is the n-th moment of a signed function v(x) on (0,27/4), i.e., in Maple notation, a(n+3) = int(x^n*v(x) , x = 0..27/4), n = 0,1..., where v(x) = -sqrt(3)*x^(4/3)*(7*x^(1/3)*hypergeom([-5/6, -1/3, 8/3], [2/3, 4/3], 4*x/27))-3*hypergeom([-7/6, -2/3, 7/3], [1/3, 2/3], 4*x/27)))/(6*Pi). The function v(x) vanishes at x = 0 and x = 27/4. In addition it has one zero in the interval between x = 0 and x = 27/4. (End)
D-finite with recurrence 2*n*(2*n+1)*(n-3)*a(n) -3*(3*n-5)*(n-1)*(3*n-4)*a(n-1)=0. - R. J. Mathar, Jul 26 2022

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A102893 Number of noncrossing trees with n edges and having degree of the root at least 2.

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A102594 Number of noncrossing trees with n edges in which no border edges emanate from the root.

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