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A161696 Number of reduced words of length n in the Weyl group B_3.

%I A161696 #22 Sep 08 2022 08:45:45
%S A161696 1,3,5,7,8,8,7,5,3,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
%T A161696 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
%U A161696 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0
%N A161696 Number of reduced words of length n in the Weyl group B_3.
%C A161696 If the zeros are ignored, this is the coordination sequence for the truncated cuboctahedron (see the Karzes link). - _N. J. A. Sloane_, Jan 08 2020
%C A161696 Computed with MAGMA using commands similar to those used to compute A161409.
%D A161696 J. E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge, 1990. See under Poincaré polynomial.
%D A161696 N. Bourbaki, Groupes et algèbres de Lie, Chap. 4, 5, 6. (The group is defined in Planche II.)
%H A161696 Tom Karzes, <a href="/A298808/a298808.html">Polyhedron Coordination Sequences</a>
%F A161696 G.f. for B_m is the polynomial Prod_{k=1..m}(1-x^(2k))/(1-x). Only finitely many terms are nonzero. This is a row of the triangle in A128084.
%p A161696 seq(coeff(series(mul((1-x^(2k))/(1-x),k=1..3),x,n+1), x, n), n = 0 .. 120); # _Muniru A Asiru_, Oct 25 2018
%t A161696 CoefficientList[Series[Product[(1-x^(2*k)), {k,1,3}] /(1-x)^3, {x,0,9}], x] (* _G. C. Greubel_, Oct 25 2018 *)
%o A161696 (PARI) t='t+O('t^10); Vec(prod(k=1,3,1-t^(2*k))/(1-t)^3) \\ _G. C. Greubel_, Oct 25 2018
%o A161696 (Magma) m:=10; R<t>:=PowerSeriesRing(Integers(), m); Coefficients(R!((&*[1-t^(2*k): k in [1..3]])/(1-t)^3)); // _G. C. Greubel_, Oct 25 2018
%Y A161696 The growth series for the finite Coxeter (or Weyl) groups B_2 through B_12 are A161696-A161699, A161716, A161717, A161733, A161755, A161776, A161858. These are all rows of A128084. The growth series for the affine Coxeter (or Weyl) groups B_2 through B_12 are A008576, A008137, A267167-A267175.
%K A161696 nonn
%O A161696 0,2
%A A161696 _John Cannon_ and _N. J. A. Sloane_, Nov 30 2009