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

%I A080721 #28 Aug 27 2021 13:04:16
%S A080721 1,1,1,1,4,4,1,9,21,14,1,16,66,100,50,1,25,160,410,455,182,1,36,330,
%T A080721 1260,2310,2016,672,1,49,609,3220,8610,12222,8778,2508,1,64,1036,7224,
%U A080721 26250,53592,61908,37752,9438,1,81,1656,14700,69300,189882,312312,303732,160875,35750,1
%N A080721 Triangle of binomial(n,k)*(binomial(n+k,k)-binomial(n+k-2,k-1)).
%C A080721 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.
%C A080721 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]
%H A080721 Michael De Vlieger, <a href="/A080721/b080721.txt">Table of n, a(n) for n = 0..11475</a> (rows 0 <= n <= 150, flattened)
%H A080721 S. Fomin and A. Zelevinsky, <a href="http://arXiv.org/abs/math/0104151">Cluster algebras I: Foundations</a>, arXiv:math/0104151 [math.RT], 2001; J. Amer. Math. Soc. 15 (2002), no. 2, 497-529.
%H A080721 S. Fomin and A. Zelevinsky, <a href="http://www.jstor.org/stable/3597238">Y-systems and generalized associahedra</a>, Ann. of Math. (2) 158 (2003), no. 3, 977-1018.
%H A080721 S. Fomin and N. Reading, <a href="https://arxiv.org/abs/math/0505518">Root systems and generalized associahedra</a>, Lecture notes for IAS/Park-City 2004; arXiv:math/0505518 [math.CO], 2005-2008. [From _Peter Bala_, Oct 28 2008]
%H A080721 Yasuaki Gyoda, <a href="https://arxiv.org/abs/2105.07974">Positive cluster complexes and tau-tilting simplicial complexes of cluster-tilted algebras of finite type</a>, arXiv:2105.07974 [math.RT], 2021, see page 34.
%e A080721 Contribution from _Peter Bala_, Oct 28 2008: (Start)
%e A080721 Triangle begins
%e A080721 n\k|..0....1....2....3....4....5
%e A080721 ================================
%e A080721 0..|..1
%e A080721 1..|..1....1
%e A080721 2..|..1....4....4
%e A080721 3..|..1....9...21...14
%e A080721 4..|..1...16...66..100...50
%e A080721 5..|..1...25..160..410..455..182
%e A080721 ...
%e A080721 (End)
%p A080721 A080721 := proc(n,k)
%p A080721     binomial(n,k)*(binomial(n+k,k)-binomial(n+k-2,k-1))
%p A080721 end proc: # _R. J. Mathar_, Mar 22 2013
%t A080721 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 *)
%o A080721 (PARI)
%o A080721 T(n,k)=binomial(n,k)*(binomial(n+k,k)-binomial(n+k-2,k-1))
%o A080721 for (n=0, 10, for (k=0,n, print1(T(n,k),", ")));
%o A080721 /* _Joerg Arndt_, Feb 21 2013 */
%Y A080721 Cf. A033282, A063007.
%Y A080721 A051924 (main diagonal), A145903( h-vectors type D associahedra). [From _Peter Bala_, Oct 28 2008]
%K A080721 easy,nonn,tabl
%O A080721 0,5
%A A080721 _Paul Boddington_, Mar 07 2003