0,2

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.

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.

Also number of signed orders.

Kenneth S. Brown, Buildings, Springer-Verlag, 1989

Peter C. Fishburn, Signed Orders, Choice Probabilities and Linear Polytopes, Journal of Mathematical Psychology, Volume 45, Issue 1, (2001), pp. 53-80.

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

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

a(n) = sum(binomial(n, t)*2^(n-t)*A000670(n-t), t=0..n); # Fishburn 2001, p. 57.

a(n) = Sum_{k=0..n} Stirling2(n, k)*k!*A001333(k+1). - Vladeta Jovovic (vladeta(AT)eunet.rs), Sep 28 2003

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]

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]

Half the row sums of triangle A154695. - Gerson W. Barbosa (gerson.w.barbosa(AT)gmail.com), Jun 26 2009

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.

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)

Cf. A000670, A080254.

Sequence in context: A140983 A138013 A052807 this_sequence A009813 A135750 A007767

Adjacent sequences: A080250 A080251 A080252 this_sequence A080254 A080255 A080256

easy,nonn

Paul Boddington and Tim Honeywill (psb(AT)maths.warwick.ac.uk), Feb 10 2003

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