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Number of positive clusters of type D_n.
6

%I #63 Sep 08 2022 08:45:30

%S -1,1,5,20,77,294,1122,4290,16445,63206,243542,940576,3640210,

%T 14115100,54826020,213286590,830905245,3241119750,12657425550,

%U 49483369320,193641552390,758454277620,2973183318300,11664026864100,45791597230002,179892016853724

%N Number of positive clusters of type D_n.

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

%C 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

%C 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

%H G. C. Greubel, <a href="/A129869/b129869.txt">Table of n, a(n) for n = 1..1000</a>

%H JL Baril, S Kirgizov, <a href="http://jl.baril.u-bourgogne.fr/Stirling.pdf">The pure descent statistic on permutations</a>, Preprint, 2016, See Cor. 6.

%H Paul Barry, <a href="https://arxiv.org/abs/2004.04577">On a Central Transform of Integer Sequences</a>, arXiv:2004.04577 [math.CO], 2020.

%H F. Chapoton and L. Manivel, <a href="http://arxiv.org/abs/1109.6490">Triangulations and Severi varieties</a>, arXiv:1109.6490 [math.AG], 2011.

%H 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 Milan Janjic, <a href="http://www.pmfbl.org/janjic/">Two Enumerative Functions</a>

%H M. A. A. Obaid, S. K. Nauman, W. M. Fakieh, C. M. Ringel, <a href="http://www.math.uni-bielefeld.de/~ringel/opus/jeddah.pdf">The numbers of support-tilting modules for a Dynkin algebra</a>, 2014 and <a href="https://cs.uwaterloo.ca/journals/JIS/VOL18/Ringel/ringel22.html">J. Int. Seq. 18 (2015) 15.10.6</a> .

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

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

%F 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

%F 2*n*a(n) +(-13*n+14)*a(n-1) +10*(2*n-5)*a(n-2)=0. - _R. J. Mathar_, Apr 11 2013

%F 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

%e 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.

%p a := n -> (1/8)*4^n*GAMMA(-1/2+n)*(3*n-4)/(sqrt(Pi)*GAMMA(1+n)) - 0^(n-1)/2;

%p seq(a(n), n=1..26); # _Peter Luschny_, Dec 14 2015

%t Table[((3*n-4)/n)*Binomial[2n-3,n-1],{n,30}] (* _Harvey P. Dale_, May 23 2012 *)

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

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

%o (Magma) [(3*n-4)/n * Binomial(2*n-3,n-1) : n in [1..30]]; // _Wesley Ivan Hurt_, Jan 24 2017

%Y Cf. A051924.

%K sign,easy

%O 1,3

%A _F. Chapoton_, May 24 2007