OFFSET
0,5
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]
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.
EXAMPLE
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)
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 *)
PROG
(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 */
CROSSREFS
KEYWORD
AUTHOR
Paul Boddington, Mar 07 2003
STATUS
approved