%I A080253
%S A080253 1,3,17,147,1697,24483,423857,8560947,197613377,5131725123,148070287697,
%T A080253 4699645934547,162723741209057,6103779096411363,246564971326084337,
%U A080253 10671541841672056947,492664975795819140737,24166020791610523843203
%N A080253 a(n) is the number of elements in the Coxeter complex of type B_n (or 
               C_n).
%C A080253 There is a nice geometric interpretation. Let V be a Euclidean space 
               containing a root system of type B_n. We can decompose V into a disjoint 
               union of 'cells', a cell being simply a maximal connected subset 
               C of V with the property that if C has nonempty intersection with 
               the orthogonal complement of some root a, then C lies entirely within 
               the orthogonal complement of a. a(n) is then the number of cells.
%C A080253 For example if n=2 then we can take V=R^2 and the roots to be (1,0), 
               (0,1), (1,1), (-1, -1) and their negatives. The 17 cells are as follows: 
               the set containing the origin O; the eight 'open' halflines radiating 
               from O and containing a root (but not O); the eight connected components 
               of V minus the union of the nine cells already described. The corresponding 
               sequences for types A,D are A000670, A080254 respectively.
%C A080253 Also number of signed orders.
%D A080253 Kenneth S. Brown, Buildings, Springer-Verlag, 1989
%D A080253 Peter C. Fishburn, Signed Orders, Choice Probabilities and Linear Polytopes, 
               Journal of Mathematical Psychology, Volume 45, Issue 1, (2001), pp. 
               53-80.
%F A080253 a(n) = 1 + sum('2^r*binomial(n, r)*a(n-r)', 'r'=1..n)
%F A080253 E.g.f: exp(x)/(2-exp(2*x)) - Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), 
               Feb 14 2003
%F A080253 a(n) = sum(binomial(n, t)*2^(n-t)*A000670(n-t), t=0..n); # Fishburn 2001, 
               p. 57.
%F A080253 a(n) = Sum_{k=0..n} Stirling2(n, k)*k!*A001333(k+1). - Vladeta Jovovic 
               (vladeta(AT)eunet.rs), Sep 28 2003
%F A080253 a(n) = 1/2 * sum('(2*k+1)^n/2^k', 'k'=0..inf) [From Gerson W. Barbosa 
               (gerson.w.barbosa(AT)gmail.com), May 11 2009]
%F A080253 An approximation formula can be derived from the latter, a(n) ~ n!/(2*sqrt2)*(2/
               ln(2))^(n+1), with relative errors approaching asymptotically zero 
               as n increases. [From Gerson W. Barbosa (gerson.w.barbosa(AT)gmail.com), 
               Jun 26 2009]
%F A080253 Half the row sums of triangle A154695. - Gerson W. Barbosa (gerson.w.barbosa(AT)gmail.com), 
               Jun 26 2009
%e A080253 a(2)=17 as follows. Let (W,S) be a Coxeter system of type B_2. By definition 
               the elements of the associated complex are right cosets of 'special 
               parabolic subgroups'. These are simply the subgroups generated by 
               subsets of S. In our case they have orders 1,2,2,8 and hence have 
               8,4,4,1 cosets respectively, giving a total of 17.
%p A080253 A080253 := proc(n) option remember; local k; if n <1 then 1 else 1 + 
               add(2^r*binomial(n,r)*A080253(n-r),r=1..n); fi; end; seq(A080253(n),
               n=0..30); (from Detlef Pauly)
%Y A080253 Cf. A000670, A080254.
%Y A080253 Sequence in context: A140983 A138013 A052807 this_sequence A009813 A135750 
               A007767
%Y A080253 Adjacent sequences: A080250 A080251 A080252 this_sequence A080254 A080255 
               A080256
%K A080253 easy,nonn
%O A080253 0,2
%A A080253 Paul Boddington and Tim Honeywill (psb(AT)maths.warwick.ac.uk), Feb 10 
               2003
%E A080253 More terms from Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), 
               Feb 14 2003