The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation. Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A129869 Number of positive clusters of type D_n. 6
 -1, 1, 5, 20, 77, 294, 1122, 4290, 16445, 63206, 243542, 940576, 3640210, 14115100, 54826020, 213286590, 830905245, 3241119750, 12657425550, 49483369320, 193641552390, 758454277620, 2973183318300, 11664026864100, 45791597230002, 179892016853724 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS This is also the number of antichains in the poset of positive-but-not-simple roots of type D_n. 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 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 LINKS G. C. Greubel, Table of n, a(n) for n = 1..1000 JL Baril, S Kirgizov, The pure descent statistic on permutations, Preprint, 2016, See Cor. 6. Paul Barry, On a Central Transform of Integer Sequences, arXiv:2004.04577 [math.CO], 2020. F. Chapoton and L. Manivel, Triangulations and Severi varieties, arXiv:1109.6490 [math.AG], 2011. S. Fomin and A. Zelevinsky, Y-systems and generalized associahedra, Ann. of Math. (2) 158 (2003), no. 3, 977-1018. Milan Janjic, Two Enumerative Functions M. A. A. Obaid, S. K. Nauman, W. M. Fakieh, C. M. Ringel, The numbers of support-tilting modules for a Dynkin algebra, 2014 and J. Int. Seq. 18 (2015) 15.10.6 . FORMULA a(n) = (3*n-4)/n * C(2*n-3,n-1). Starting with "1" = the Narayana transform (A001263) of [1, 4, 7, 10, 13, 16, ...]. - Gary W. Adamson, Jul 29 2001 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 2*n*a(n) +(-13*n+14)*a(n-1) +10*(2*n-5)*a(n-2)=0. - R. J. Mathar, Apr 11 2013 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 EXAMPLE 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. MAPLE a := n -> (1/8)*4^n*GAMMA(-1/2+n)*(3*n-4)/(sqrt(Pi)*GAMMA(1+n)) - 0^(n-1)/2; seq(a(n), n=1..26); # Peter Luschny, Dec 14 2015 MATHEMATICA Table[((3*n-4)/n)*Binomial[2n-3, n-1], {n, 30}] (* Harvey P. Dale, May 23 2012 *) PROG (MuPAD) (3*n-4)/n*binomial(2*n-3, n-1) \$n=1..22; (Sage) [(3*n-4)/n*binomial(2*n-3, n-1) for n in range(1, 20)] (MAGMA) [(3*n-4)/n * Binomial(2*n-3, n-1) : n in [1..30]]; // Wesley Ivan Hurt, Jan 24 2017 CROSSREFS Cf. A051924. Sequence in context: A295347 A270985 A289786 * A271887 A079737 A028814 Adjacent sequences:  A129866 A129867 A129868 * A129870 A129871 A129872 KEYWORD sign,easy,changed AUTHOR F. Chapoton, May 24 2007 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified August 6 15:50 EDT 2020. Contains 336255 sequences. (Running on oeis4.)