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A080253 a(n) is the number of elements in the Coxeter complex of type B_n (or C_n). 18


%S 1,3,17,147,1697,24483,423857,8560947,197613377,5131725123,

%T 148070287697,4699645934547,162723741209057,6103779096411363,

%U 246564971326084337,10671541841672056947,492664975795819140737,24166020791610523843203

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

%C 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 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 Also number of signed orders.

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

%H Vincenzo Librandi, <a href="/A080253/b080253.txt">Table of n, a(n) for n = 0..200</a>

%H 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 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 Joël Gay, Vincent Pilaud, <a href="https://arxiv.org/abs/1804.06572">The weak order on Weyl posets</a>, arXiv:1804.06572 [math.CO], 2018.

%H Eric Weisstein's MathWorld, <a href="http://mathworld.wolfram.com/Polylogarithm.html">Polylogarithm</a>.

%F a(n) = 1 + sum_{r=1..n} 2^r *binomial(n, r) *a(n-r).

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

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

%F a(n) = Sum_{k=0..n} Stirling2(n, k)*k!*A001333(k+1). - _Vladeta Jovovic_, Sep 28 2003

%F 2*a(n) = sum_{k=0..infinity} (2*k+1)^n/2^k = 2^n*LerchPhi(1/2,-n,1/2). - _Gerson Washiski Barbosa_, May 11 2009, Dec 12 2010

%F 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 Half the row sums of triangle A154695. - _Gerson Washiski Barbosa_, Jun 26 2009

%F 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 to due Stieltjes). - _Sergei N. Gladkovskii_, Jan 11 2013

%F a(n) = sum_{k = 0..n} A060187(n, k)*2^(n-k). - _Peter Luschny_, Apr 26 2013

%F 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 a(n) = log(2) * int {x = 0..inf} (2*floor(x) + 1)^n * 2^(-x) dx. - _Peter Bala_, Feb 06 2015

%F From _Vladimir Reshetnikov_, Oct 31 2015: (Start)

%F a(n) = (-1)^(n+1)*(Li_{-n}(sqrt(2)) - Li_{-n}(-sqrt(2)))/(2*sqrt(2)), where Li_n(x) is the polylogarithm.

%F Li_{-n}(sqrt(2)) = (-1)^(n+1)*(2*A216794(n) + a(n)*sqrt(2)).

%F (End)

%e 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 := 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 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 CoefficientList[Series[E^x/(2-E^(2*x)), {x, 0, 20}], x] * Range[0, 20]! (* _Vaclav Kotesovec_, Feb 07 2015 *)

%t 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 (Maxima) t(n):=sum(stirling2(n,k)*k!,k,0,n);

%o c(n):=sum(binomial(n,k)*2^k*t(k),k,0,n);

%o makelist(c(n),n,0,40); // _Emanuele Munarini_, Oct 04 2012

%o (Sage)

%o def A080253(n):

%o return add(A060187(n, k) << (n-k) for k in (0..n))

%o [A080253(n) for n in (0..17)] # _Peter Luschny_, Apr 26 2013

%Y Cf. A000670, A080254, A216794.

%K easy,nonn

%O 0,2

%A _Paul Boddington_ and _Tim Honeywill_, Feb 10 2003

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

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Last modified February 17 02:33 EST 2019. Contains 320200 sequences. (Running on oeis4.)