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A080253 a(n) is the number of elements in the Coxeter complex of type B_n (or C_n). 28
1, 3, 17, 147, 1697, 24483, 423857, 8560947, 197613377, 5131725123, 148070287697, 4699645934547, 162723741209057, 6103779096411363, 246564971326084337, 10671541841672056947, 492664975795819140737, 24166020791610523843203 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
COMMENTS
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.
REFERENCES
Kenneth S. Brown, Buildings, Springer-Verlag, 1989.
LINKS
Paul Barry, General Eulerian Polynomials as Moments Using Exponential Riordan Arrays, Journal of Integer Sequences, 16 (2013), #13.9.6.
Peter C. Fishburn, Signed Orders, Choice Probabilities and Linear Polytopes, Journal of Mathematical Psychology, Volume 45, Issue 1, (2001), pp. 53-80.
Joël Gay and Vincent Pilaud, The weak order on Weyl posets, arXiv:1804.06572 [math.CO], 2018.
Eric Weisstein's MathWorld, Polylogarithm.
FORMULA
a(n) = 1 + Sum_{r=1..n} 2^r *binomial(n, r) *a(n-r).
E.g.f.: exp(x)/(2-exp(2*x)). - Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Feb 14 2003
a(n) = Sum_{t=0..n} binomial(n, t)*2^(n-t)*A000670(n-t). Fishburn 2001, p. 57.
a(n) = Sum_{k=0..n} Stirling2(n, k)*k!*A001333(k+1). - Vladeta Jovovic, Sep 28 2003
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
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
Half the row sums of triangle A154695. - Gerson Washiski Barbosa, Jun 26 2009
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
a(n) = Sum_{k = 0..n} A060187(n, k)*2^(n-k). - Peter Luschny, Apr 26 2013
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
a(n) = log(2) * Integral_{x = 0..oo} (2*floor(x) + 1)^n * 2^(-x) dx. - Peter Bala, Feb 06 2015
From Vladimir Reshetnikov, Oct 31 2015: (Start)
a(n) = (-1)^(n+1)*(Li_{-n}(sqrt(2)) - Li_{-n}(-sqrt(2)))/(2*sqrt(2)), where Li_n(x) is the polylogarithm.
Li_{-n}(sqrt(2)) = (-1)^(n+1)*(2*A216794(n) + a(n)*sqrt(2)).
(End)
EXAMPLE
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.
MAPLE
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
MATHEMATICA
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 *)
CoefficientList[Series[E^x/(2-E^(2*x)), {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Feb 07 2015 *)
Round@Table[(-1)^(n + 1) (PolyLog[-n, Sqrt[2]] - PolyLog[-n, -Sqrt[2]])/(2 Sqrt[2]), {n, 0, 20}] (* Vladimir Reshetnikov, Oct 31 2015 *)
PROG
(Maxima) t(n):=sum(stirling2(n, k)*k!, k, 0, n);
c(n):=sum(binomial(n, k)*2^k*t(k), k, 0, n);
makelist(c(n), n, 0, 40); // Emanuele Munarini, Oct 04 2012
(Sage)
def A080253(n):
return add(A060187(n, k) << (n-k) for k in (0..n))
[A080253(n) for n in (0..17)] # Peter Luschny, Apr 26 2013
CROSSREFS
Sequence in context: A277466 A138013 A052807 * A234289 A009813 A319946
KEYWORD
easy,nonn
AUTHOR
EXTENSIONS
More terms from Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Feb 14 2003
STATUS
approved

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Last modified June 16 08:45 EDT 2024. Contains 373424 sequences. (Running on oeis4.)