0,5

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.

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]

K. S. Brown, Buildings, Springer-Verlag, 1988

C. Chow, On the Eulerian polynomials of type D.

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).

Chow gives complicated recurrences and generating functions.

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]

Contribution from Peter Bala (pbala(AT)toucansurf.com), Oct 29 2008: (Start)

The triangle begins

n\k|..0....1....2....3....4....5

================================

0..|..1

1..|..1....1

2..|..1....2....1

3..|..1...11...11....1

4..|..1...44..102...44....1

5..|..1..157..802..802..157....1

...

(End)

Cf. A008292, A060187, A080254.

A145902. [From Peter Bala (pbala(AT)toucansurf.com), Oct 29 2008]

Sequence in context: A165883 A110905 A158202 this_sequence A010246 A156885 A054505

Adjacent sequences: A066091 A066092 A066093 this_sequence A066095 A066096 A066097

easy,nonn,tabl

Paul Boddington (psb(AT)maths.warwick.ac.uk), Mar 05 2003