%I A066094
%S A066094 1,1,1,1,2,1,1,11,11,1,1,44,102,44,1,1,157,802,802,157,1,1,530,5551,
%T A066094 10876,5551,530,1,1,1731,35121,124427,124427,35121,1731,1,1,5528,208732,
%U A066094 1265704,2201030,1265704,208732,5528,1
%N A066094 Type D Eulerian triangle.
%C A066094 Let n >= 2 and write the polynomial D(n,0)+D(n,1)*x+...+D(n,n)*x^n as 
               a polynomial in y := x-1. Then the coefficient of y^r is the number 
               of cells of dimension n-r in the cellular decomposition of a Euclidean 
               space containing a root system of type D_n. If n >= 2 then the corresponding 
               row sum is 2^(n-1)*(n-1)!, while sum(2^k*D(n,k),k=0..n) is given 
               by sequence A080254.
%C A066094 The entries in row n (for n >= 2) are the components of the h-vector 
               of the permutohedra of type D_n. See A145902 for the corresponding 
               array of f-vectors for type D permutohedra. [From Peter Bala (pbala(AT)toucansurf.com), 
               Oct 29 2008]
%D A066094 K. S. Brown, Buildings, Springer-Verlag, 1988
%H A066094 C. Chow, <a href="http://arXiv.org/abs/math.CO/0201140">On the Eulerian 
               polynomials of type D</a>.
%F A066094 Let D(n, k) denote the (k+1)st entry in the (n+1)st row and let A(n, 
               k), B(n, k) be triangles A008292 (The Eulerian triangle), A060187 
               respectively. Then D(n, k)=B(n, k)-2^(n-1)*n*A(n-2, k-1).
%F A066094 Chow gives complicated recurrences and generating functions.
%F A066094 E.g.f.: [(1-x)*exp(z*(1-x)) - z*x*(1-x)*exp(2*z*(1-x))]/(1 - x*exp(2*z*(1-x))) 
               = 1 + x*z + (1 + 2*x + x^2)*z^2/2! + (1 + 11*x + 11*x^2 + x^3)*z^3/
               3! + ... . [From Peter Bala (pbala(AT)toucansurf.com), Oct 29 2008]
%e A066094 Contribution from Peter Bala (pbala(AT)toucansurf.com), Oct 29 2008: 
               (Start)
%e A066094 The triangle begins
%e A066094 n\k|..0....1....2....3....4....5
%e A066094 ================================
%e A066094 0..|..1
%e A066094 1..|..1....1
%e A066094 2..|..1....2....1
%e A066094 3..|..1...11...11....1
%e A066094 4..|..1...44..102...44....1
%e A066094 5..|..1..157..802..802..157....1
%e A066094 ...
%e A066094 (End)
%Y A066094 Cf. A008292, A060187, A080254.
%Y A066094 A145902. [From Peter Bala (pbala(AT)toucansurf.com), Oct 29 2008]
%Y A066094 Sequence in context: A165883 A110905 A158202 this_sequence A010246 A156885 
               A054505
%Y A066094 Adjacent sequences: A066091 A066092 A066093 this_sequence A066095 A066096 
               A066097
%K A066094 easy,nonn,tabl
%O A066094 0,5
%A A066094 Paul Boddington (psb(AT)maths.warwick.ac.uk), Mar 05 2003