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A129869
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Number of positive clusters of type D_n.
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6
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-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
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OFFSET
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1,3
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COMMENTS
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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
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LINKS
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FORMULA
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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
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EXAMPLE
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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.
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MAPLE
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a := n -> (1/8)*4^n*GAMMA(-1/2+n)*(3*n-4)/(sqrt(Pi)*GAMMA(1+n)) - 0^(n-1)/2;
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MATHEMATICA
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Table[((3*n-4)/n)*Binomial[2n-3, n-1], {n, 30}] (* Harvey P. Dale, May 23 2012 *)
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PROG
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(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
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CROSSREFS
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KEYWORD
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sign,easy
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AUTHOR
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STATUS
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approved
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