|
| |
|
|
A066094
|
|
Type D Eulerian triangle.
|
|
8
| |
|
|
1, 1, 1, 1, 2, 1, 1, 11, 11, 1, 1, 44, 102, 44, 1, 1, 157, 802, 802, 157, 1, 1, 530, 5551, 10876, 5551, 530, 1, 1, 1731, 35121, 124427, 124427, 35121, 1731, 1, 1, 5528, 208732, 1265704, 2201030, 1265704, 208732, 5528, 1
(list; table; graph; refs; listen; history; internal format)
|
|
|
|
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-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]
|
|
|
REFERENCES
| K. S. Brown, Buildings, Springer-Verlag, 1988
|
|
|
LINKS
| C. Chow, On the Eulerian polynomials of type D.
|
|
|
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! + ... . [From Peter Bala (pbala(AT)toucansurf.com), Oct 29 2008]
|
|
|
EXAMPLE
| 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)
|
|
|
CROSSREFS
| Cf. A008292, A060187, A080254.
A145902. [From Peter Bala (pbala(AT)toucansurf.com), Oct 29 2008]
Sequence in context: A158202 A176307 A197648 * A160625 A010246 A186430
Adjacent sequences: A066091 A066092 A066093 * A066095 A066096 A066097
|
|
|
KEYWORD
| easy,nonn,tabl
|
|
|
AUTHOR
| Paul Boddington (psb(AT)maths.warwick.ac.uk), Mar 05 2003
|
| |
|
|
