A102594 Number of noncrossing trees with n edges in which no border edges emanate from the root.
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
Keywords
Examples
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.
Links
- 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.
Programs
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Maple
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);
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Mathematica
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 *) -
PARI
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
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PARI
Vec((g->g*(1+x-2*x*g))(1+serreverse(x/(1+x)^3 + O(x^30)))) \\ Andrew Howroyd, Nov 17 2017
Formula
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