OFFSET
0,5
COMMENTS
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!, while Sum_{k=0..n} 2^k*D(n,k) is given by sequence A080254. [Row sum formula corrected by Joshua Swanson, Jul 12 2022]
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. [Peter Bala, Oct 29 2008]
REFERENCES
K. S. Brown, Buildings, Springer-Verlag, 1988
T. K. Petersen, Eulerian Numbers, Birkhauser, 2015, Chapter 11.
LINKS
Jose Bastidas, Table of n, a(n) for n = 0..1274 (First 50 rows)
Anna Borowiec and Wojciech Mlotkowski, New Eulerian numbers of type D, arXiv:1509.03758 [math.CO], 2015.
C. Chow, On the Eulerian polynomials of type D, arXiv:math/0201140 [math.CO], 2002.
S.-M. Ma, Polynomials with only real zeros and the Eulerian polynomials of type D, arXiv preprint arXiv:1205.6242 [math.CO], 2012. - From N. J. A. Sloane, Oct 23 2012
FORMULA
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! + ... . [Peter Bala, Oct 29 2008]
EXAMPLE
From Peter Bala, 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)
CROSSREFS
KEYWORD
AUTHOR
Paul Boddington, Mar 05 2003
STATUS
approved