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A080253 a(n) is the number of elements in the Coxeter complex of type B_n (or C_n).

%I A080253 #73 Feb 16 2025 08:32:48
%S A080253 1,3,17,147,1697,24483,423857,8560947,197613377,5131725123,
%T A080253 148070287697,4699645934547,162723741209057,6103779096411363,
%U A080253 246564971326084337,10671541841672056947,492664975795819140737,24166020791610523843203
%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.
%H A080253 Vincenzo Librandi, <a href="/A080253/b080253.txt">Table of n, a(n) for n = 0..200</a>
%H A080253 Paul Barry, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL16/Barry4/barry271.html">General Eulerian Polynomials as Moments Using Exponential Riordan Arrays</a>, Journal of Integer Sequences, 16 (2013), #13.9.6.
%H A080253 Peter C. Fishburn, <a href="http://dx.doi.org/10.1006/jmps.1999.1288">Signed Orders, Choice Probabilities and Linear Polytopes</a>, Journal of Mathematical Psychology, Volume 45, Issue 1, (2001), pp. 53-80.
%H A080253 Joël Gay and Vincent Pilaud, <a href="https://arxiv.org/abs/1804.06572">The weak order on Weyl posets</a>, arXiv:1804.06572 [math.CO], 2018.
%H A080253 Eric Weisstein's MathWorld, <a href="https://mathworld.wolfram.com/Polylogarithm.html">Polylogarithm</a>.
%F A080253 a(n) = 1 + Sum_{r=1..n} 2^r *binomial(n, r) *a(n-r).
%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_{t=0..n} binomial(n, t)*2^(n-t)*A000670(n-t). Fishburn 2001, p. 57.
%F A080253 a(n) = Sum_{k=0..n} Stirling2(n, k)*k!*A001333(k+1). - _Vladeta Jovovic_, Sep 28 2003
%F A080253 2*a(n) = Sum_{k>=0} (2*k+1)^n/2^k = 2^n*LerchPhi(1/2,-n,1/2). - _Gerson Washiski Barbosa_, May 11 2009, Dec 12 2010
%F A080253 An approximation formula can be derived from the latter, a(n) ~ (n!/(2*sqrt(2)))*(2/log(2))^(n+1), with relative errors approaching asymptotically zero as n increases. - _Gerson Washiski Barbosa_, Jun 26 2009
%F A080253 Half the row sums of triangle A154695. - _Gerson Washiski Barbosa_, Jun 26 2009
%F A080253 G.f.: 1 + x/G(0) where G(k) = 1 - x*3*(2*k+1) + x^2*(k+1)*(k+1)*(1-3^2)/G(k+1); (continued fraction due to Stieltjes). - _Sergei N. Gladkovskii_, Jan 11 2013
%F A080253 a(n) = Sum_{k = 0..n} A060187(n, k)*2^(n-k). - _Peter Luschny_, Apr 26 2013
%F A080253 G.f.: 1/Q(0), where Q(k) = 1 - 3*x*(2*k+1) - 8*x^2*(k+1)^2/Q(k+1); (continued fraction). - _Sergei N. Gladkovskii_, Sep 28 2013
%F A080253 a(n) = log(2) * Integral_{x = 0..oo} (2*floor(x) + 1)^n * 2^(-x) dx. - _Peter Bala_, Feb 06 2015
%F A080253 From _Vladimir Reshetnikov_, Oct 31 2015: (Start)
%F A080253 a(n) = (-1)^(n+1)*(Li_{-n}(sqrt(2)) - Li_{-n}(-sqrt(2)))/(2*sqrt(2)), where Li_n(x) is the polylogarithm.
%F A080253 Li_{-n}(sqrt(2)) = (-1)^(n+1)*(2*A216794(n) + a(n)*sqrt(2)).
%F A080253 (End)
%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); # Detlef Pauly
%t A080253 t[n_] := Sum[StirlingS2[n, k] k!, {k, 0, n}]; c[n_] := Sum[Binomial[n, k] 2^k t[k], {k, 0, n}]; Table[c[n], {n, 0, 100}] (* _Emanuele Munarini_, Oct 04 2012 *)
%t A080253 CoefficientList[Series[E^x/(2-E^(2*x)), {x, 0, 20}], x] * Range[0, 20]! (* _Vaclav Kotesovec_, Feb 07 2015 *)
%t A080253 Round@Table[(-1)^(n + 1) (PolyLog[-n, Sqrt[2]] - PolyLog[-n, -Sqrt[2]])/(2 Sqrt[2]), {n, 0, 20}] (* _Vladimir Reshetnikov_, Oct 31 2015 *)
%o A080253 (Maxima) t(n):=sum(stirling2(n,k)*k!,k,0,n);
%o A080253 c(n):=sum(binomial(n,k)*2^k*t(k),k,0,n);
%o A080253 makelist(c(n),n,0,40); // _Emanuele Munarini_, Oct 04 2012
%o A080253 (Sage)
%o A080253 def A080253(n):
%o A080253     return add(A060187(n, k) << (n-k) for k in (0..n))
%o A080253 [A080253(n) for n in (0..17)]  # _Peter Luschny_, Apr 26 2013
%Y A080253 Cf. A000670, A080254, A216794.
%K A080253 easy,nonn
%O A080253 0,2
%A A080253 _Paul Boddington_ and _Tim Honeywill_, Feb 10 2003
%E A080253 More terms from Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Feb 14 2003