A080254 For n>3, a(n) is the number of elements in the Coxeter complex of type D_n (although the sequence starts at n=0. See comments below for precise explanation).

1, 1, 9, 75, 865, 12483, 216113, 4364979, 100757313, 2616517443, 75496735057, 2396212835283, 82968104980961, 3112139513814243, 125716310807844081, 5441108944839913587, 251195548533025953409, 12321551453507301079683

Offset

0,3

Comments

The sequence makes most sense when n>3. The values for a(2) and a(3) make sense if we regard D_2=A_1 x A_1 and D_3=A_3. The values for a(0) and a(1) have to be regarded as conventions and were included to give a nice recursive description. The corresponding sequence for type B is A080253. There one can find a worked example as well as a geometric interpretation.

Also, Eulerian D-polynomials (A066094) evaluated at 2. - Ralf Stephan, Apr 23 2004

References

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

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Joël Gay, Vincent Pilaud, The weak order on Weyl posets, arXiv:1804.06572 [math.CO], 2018.

Formula

a(0)=a(1)=1. For n>1, a(n)=1 + sum('2^r*binomial(n, r)*a(n-r)', 'r'=1..n)

E.g.f: (2*x-exp(x))/(exp(2*x)-2) - Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Feb 14 2003

a(n) ~ n! * (sqrt(2)/log(2)-1)/2 * (2/log(2))^n. - Vaclav Kotesovec, Oct 08 2013

Mathematica

CoefficientList[Series[(2*x-E^x)/(E^(2*x)-2), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Oct 08 2013 *)

Crossrefs

Cf. A000670, A080253.

Sequence in context: A335345 A126965 A066222 * A190916 A161736 A226180

Adjacent sequences: A080251 A080252 A080253 * A080255 A080256 A080257

Keyword

easy,nonn

Author

Paul Boddington & Tim Honeywill, Feb 10 2003

Extensions

More terms from Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Feb 14 2003

Status

approved