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A129869 Number of positive clusters of type D_n.

%I A129869 #63 Sep 08 2022 08:45:30
%S A129869 -1,1,5,20,77,294,1122,4290,16445,63206,243542,940576,3640210,
%T A129869 14115100,54826020,213286590,830905245,3241119750,12657425550,
%U A129869 49483369320,193641552390,758454277620,2973183318300,11664026864100,45791597230002,179892016853724
%N A129869 Number of positive clusters of type D_n.
%C A129869 This is also the number of antichains in the poset of positive-but-not-simple roots of type D_n.
%C A129869 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 A129869 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 A129869 G. C. Greubel, <a href="/A129869/b129869.txt">Table of n, a(n) for n = 1..1000</a>
%H A129869 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 A129869 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 A129869 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 A129869 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 A129869 Milan Janjic, <a href="http://www.pmfbl.org/janjic/">Two Enumerative Functions</a>
%H A129869 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 A129869 a(n) = (3*n-4)/n * C(2*n-3,n-1).
%F A129869 Starting with "1" = the Narayana transform (A001263) of [1, 4, 7, 10, 13, 16, ...]. - _Gary W. Adamson_, Jul 29 2001
%F A129869 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 A129869 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 A129869 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 A129869 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 A129869 a := n -> (1/8)*4^n*GAMMA(-1/2+n)*(3*n-4)/(sqrt(Pi)*GAMMA(1+n)) - 0^(n-1)/2;
%p A129869 seq(a(n), n=1..26); # _Peter Luschny_, Dec 14 2015
%t A129869 Table[((3*n-4)/n)*Binomial[2n-3,n-1],{n,30}] (* _Harvey P. Dale_, May 23 2012 *)
%o A129869 (MuPAD) (3*n-4)/n*binomial(2*n-3,n-1) $n=1..22;
%o A129869 (Sage) [(3*n-4)/n*binomial(2*n-3,n-1) for n in range(1,20)]
%o A129869 (Magma) [(3*n-4)/n * Binomial(2*n-3,n-1) : n in [1..30]]; // _Wesley Ivan Hurt_, Jan 24 2017
%Y A129869 Cf. A051924.
%K A129869 sign,easy
%O A129869 1,3
%A A129869 _F. Chapoton_, May 24 2007