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A145903 Generalized Narayana numbers for root systems of type D_n. Triangle of h-vectors of type D associahedra. 3
1, 1, 1, 1, 2, 1, 1, 6, 6, 1, 1, 12, 24, 12, 1, 1, 20, 70, 70, 20, 1, 1, 30, 165, 280, 165, 30, 1, 1, 42, 336, 875, 875, 336, 42, 1, 1, 56, 616, 2296, 3500, 2296, 616, 56, 1, 1, 72, 1044, 5292, 11466, 11466, 5292, 1044, 72, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

The generalized Narayana numbers of type D_n (row n of this triangle) are defined as the entries of the h-vector of the simplicial complex dual to the generalized associahedron of type D_n [Fomin & Reading, p.60]. For the corresponding triangle of f-vectors see A080721. For Narayana numbers of root systems of type A and type B see A001263 and A008459 respectively.

REFERENCES

T. K. Petersen, Eulerian Numbers, Birkhauser, 2015, Chapter 12.

LINKS

Table of n, a(n) for n=0..54.

S. Fomin, N. Reading, Root systems and generalized associahedra, Lecture notes for IAS/Park-City 2004, arXiv:math/0505518 [math.CO], 2005-2008.

FORMULA

For n >= 2, T(n,k) = binomial(n,k)^2 - n/(n-1)*binomial(n-1,k-1)*binomial(n-1,k).

EXAMPLE

Root systems of type D_n are defined only for n >= 2. It seems convenient to complete the array to form a lower unit triangular matrix.

Triangle starts

n\k|..0....1....2....3....4....5....6

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

0..|..1

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

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

3..|..1....6....6....1

4..|..1...12...24...12....1

5..|..1...20...70...70...20....1

6..|..1...30..165..280..165...30....1

...

CROSSREFS

A001263, A008459, A051924 (row sums), A080721.

Sequence in context: A157635 A075798 A155864 * A223257 A173881 A172373

Adjacent sequences:  A145900 A145901 A145902 * A145904 A145905 A145906

KEYWORD

easy,nonn,tabl

AUTHOR

Peter Bala, Oct 28 2008

STATUS

approved

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Last modified February 17 18:25 EST 2018. Contains 299296 sequences. (Running on oeis4.)