A080253 a(n) is the number of elements in the Coxeter complex of type B_n (or C_n).
1, 3, 17, 147, 1697, 24483, 423857, 8560947, 197613377, 5131725123, 148070287697, 4699645934547, 162723741209057, 6103779096411363, 246564971326084337, 10671541841672056947, 492664975795819140737, 24166020791610523843203
Offset: 0
Examples
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.
References
- Kenneth S. Brown, Buildings, Springer-Verlag, 1989.
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..200
- 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.
Programs
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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
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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 *) -
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
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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
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)
Extensions
More terms from Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Feb 14 2003
Comments