A080721 Triangle of binomial(n,k)*(binomial(n+k,k)-binomial(n+k-2,k-1)).

1, 1, 1, 1, 4, 4, 1, 9, 21, 14, 1, 16, 66, 100, 50, 1, 25, 160, 410, 455, 182, 1, 36, 330, 1260, 2310, 2016, 672, 1, 49, 609, 3220, 8610, 12222, 8778, 2508, 1, 64, 1036, 7224, 26250, 53592, 61908, 37752, 9438, 1, 81, 1656, 14700, 69300, 189882, 312312, 303732, 160875, 35750, 1

Offset: 0

Author

Paul Boddington, Mar 07 2003

Keywords

• easy
• nonn
• tabl

Comments

For n>1 and 0 <= k <= n, a(n,k) is the number of compatible k-sets of cluster variables in Fomin and Zelevinsky's 'cluster algebra' of finite type D_n.

Triangle of f-vectors of the simplicial complexes dual to the generalized associahedra of type D_n (n >= 2). See A145903 for the corresponding triangle of h-vectors. For the triangles of f-vectors of type A and type B associahedra see A033282 and A063007 respectively. [Peter Bala, Oct 28 2008]

Examples

			Contribution from _Peter Bala_, Oct 28 2008: (Start)
Triangle begins
n\k|..0....1....2....3....4....5
================================
0..|..1
1..|..1....1
2..|..1....4....4
3..|..1....9...21...14
4..|..1...16...66..100...50
5..|..1...25..160..410..455..182
...
(End)
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 0..11475 (rows 0 <= n <= 150, flattened)
• S. Fomin and A. Zelevinsky, Cluster algebras I: Foundations, arXiv:math/0104151 [math.RT], 2001; J. Amer. Math. Soc. 15 (2002), no. 2, 497-529.
• S. Fomin and A. Zelevinsky, Y-systems and generalized associahedra, Ann. of Math. (2) 158 (2003), no. 3, 977-1018.
• S. Fomin and N. Reading, Root systems and generalized associahedra, Lecture notes for IAS/Park-City 2004; arXiv:math/0505518 [math.CO], 2005-2008. [From _Peter Bala_, Oct 28 2008]
• Yasuaki Gyoda, Positive cluster complexes and tau-tilting simplicial complexes of cluster-tilted algebras of finite type, arXiv:2105.07974 [math.RT], 2021, see page 34.

Crossrefs

Cf. A033282, A063007.

A051924 (main diagonal), A145903( h-vectors type D associahedra). [From Peter Bala, Oct 28 2008]

Programs

Maple

A080721 := proc(n,k)
    binomial(n,k)*(binomial(n+k,k)-binomial(n+k-2,k-1))
end proc: # R. J. Mathar, Mar 22 2013

Mathematica

Flatten[Table[Binomial[n,k](Binomial[n+k,k]-Binomial[Abs[n+k-2],k-1]),{n,0,10},{k,0,n}]] (* Harvey P. Dale, Feb 20 2013 *)

PARI

T(n,k)=binomial(n,k)*(binomial(n+k,k)-binomial(n+k-2,k-1))
for (n=0, 10, for (k=0,n, print1(T(n,k),", ")));
/* Joerg Arndt, Feb 21 2013 */