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A102893
Number of noncrossing trees with n edges and having degree of the root at least 2.
9
1, 0, 1, 5, 25, 130, 700, 3876, 21945, 126500, 740025, 4382625, 26225628, 158331880, 963250600, 5899491640, 36345082425, 225082957512, 1400431689475, 8749779798375, 54874635255825, 345329274848250, 2179969531405680
OFFSET
0,4
COMMENTS
[a(n+2)]= [1,5,25,130,700,...] is the self-convolution 5th power of A001764. - Philippe Deléham, Nov 11 2009
LINKS
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.
FORMULA
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
EXAMPLE
a(2)=1 because among the noncrossing trees with 2 edges, namely /_, _\ and /\, only the last one has root degree >1.
MAPLE
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
MATHEMATICA
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 *)
PROG
(PARI) a(n) = if(n<=1, n==0, 5*binomial(3*n-1, n-2)/(3*n-1)); \\ Andrew Howroyd, Nov 17 2017
CROSSREFS
Column k=0 of A102892 and column k=0 of A102593.
Sequence in context: A182626 A351050 A184139 * A094602 A207834 A351187
KEYWORD
nonn,easy
AUTHOR
Emeric Deutsch, Jan 16 2005
STATUS
approved