A129869 Number of positive clusters of type D_n.

-1, 1, 5, 20, 77, 294, 1122, 4290, 16445, 63206, 243542, 940576, 3640210, 14115100, 54826020, 213286590, 830905245, 3241119750, 12657425550, 49483369320, 193641552390, 758454277620, 2973183318300, 11664026864100, 45791597230002, 179892016853724

Offset: 1

Author

F. Chapoton, May 24 2007

Keywords

• sign
• easy

Comments

This is also the number of antichains in the poset of positive-but-not-simple roots of type D_n.

If Y is a fixed 2-subset of a (2n+1)-set X then a(n+1) is the number of (n+2)-subsets of X intersecting Y. - Milan Janjic, Oct 28 2007

Define an array by m(1,k)=k, m(n,1) = n*(n-1) + 1, and m(n,k) = m(n,k-1) + m(n-1,k) otherwise. This yields m(n,1) = A002061(n) and on the diagonal, m(n,n) = a(n+1). - J. M. Bergot, Mar 30 2013

Examples

			a(3) = 5 because the type D3 is the same as type A3 and there are 5 positive clusters among the 14 clusters in type A3.
		

Links

• G. C. Greubel, Table of n, a(n) for n = 1..1000
• JL Baril, S Kirgizov, The pure descent statistic on permutations, Preprint, 2016, See Cor. 6.
• Paul Barry, On a Central Transform of Integer Sequences, arXiv:2004.04577 [math.CO], 2020.
• F. Chapoton and L. Manivel, Triangulations and Severi varieties, arXiv:1109.6490 [math.AG], 2011.
• S. Fomin and A. Zelevinsky, Y-systems and generalized associahedra, Ann. of Math. (2) 158 (2003), no. 3, 977-1018.
• Milan Janjic, Two Enumerative Functions
• M. A. A. Obaid, S. K. Nauman, W. M. Fakieh, C. M. Ringel, The numbers of support-tilting modules for a Dynkin algebra, 2014 and J. Int. Seq. 18 (2015) 15.10.6 .

Crossrefs

Cf. A051924.

Programs

Magma

[(3*n-4)/n * Binomial(2*n-3,n-1) : n in [1..30]]; // Wesley Ivan Hurt, Jan 24 2017

Maple

a := n -> (1/8)*4^n*GAMMA(-1/2+n)*(3*n-4)/(sqrt(Pi)*GAMMA(1+n)) - 0^(n-1)/2;
seq(a(n), n=1..26); # Peter Luschny, Dec 14 2015

Mathematica

Table[((3*n-4)/n)*Binomial[2n-3,n-1],{n,30}] (* Harvey P. Dale, May 23 2012 *)

MuPAD

(3*n-4)/n*binomial(2*n-3,n-1) $n=1..22;

Sage

[(3*n-4)/n*binomial(2*n-3,n-1) for n in range(1,20)]

Formula

a(n) = (3*n-4)/n * C(2*n-3,n-1).

Starting with "1" = the Narayana transform (A001263) of [1, 4, 7, 10, 13, 16, ...]. - Gary W. Adamson, Jul 29 2001

G.f.: x^2*(sqrt(1-4*x)*(2*x+1)-4*x+1)/(sqrt(1-4*x)*(4*x^2-5*x+1) +12*x^2-7*x+1)-x. - Vladimir Kruchinin, Sep 27 2011

2*n*a(n) +(-13*n+14)*a(n-1) +10*(2*n-5)*a(n-2)=0. - R. J. Mathar, Apr 11 2013

a(n) = (1/8)*4^n*Gamma(n-1/2)*(3*n-4)/(sqrt(Pi)*Gamma(1+n)) - 0^(n-1)/2. - Peter Luschny, Dec 14 2015