%I #28 Sep 08 2022 08:45:45
%S 1,9,44,156,449,1113,2463,4983,9372,16588,27886,44846,69387,103763,
%T 150538,212538,292779,394371,520399,673783,857121,1072521,1321430,
%U 1604470,1921291,2270451,2649332,3054100,3479715,3919995,4367735
%N Number of reduced words of length n in the Weyl group B_9.
%C Computed with MAGMA using commands similar to those used to compute A161409.
%D J. E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge, 1990. See under Poincaré polynomial.
%D N. Bourbaki, Groupes et algèbres de Lie, Chap. 4, 5, 6. (The group is defined in Planche II.)
%H G. C. Greubel, <a href="/A161733/b161733.txt">Table of n, a(n) for n = 0..81</a>
%F G.f. for B_m is the polynomial Product_{k=1..m} (1-x^(2k))/(1-x). Only finitely many terms are nonzero. This is a row of the triangle in A128084.
%p seq(coeff(series(mul((1-x^(2k))/(1-x),k=1..9),x,n+1), x, n), n = 0 .. 30); # _Muniru A Asiru_, Oct 25 2018
%t CoefficientList[Series[(1 - x^2) (1 -x^4) (1 - x^6) (1 - x^8) (1 - x^10) (1 - x^12) (1 - x^14) (1 - x^16) (1 - x^18) / (1 - x)^9, {x, 0, 81}], x] (* _Vincenzo Librandi_, Aug 22 2016 *)
%o (PARI) t='t+O('t^40); Vec(prod(k=1,9,1-t^(2*k))/(1-t)^9) \\ _G. C. Greubel_, Oct 25 2018
%o (Magma) m:=40; R<t>:=PowerSeriesRing(Integers(), m); Coefficients(R!((&*[1-t^(2*k): k in [1..9]])/(1-t)^9)); // _G. C. Greubel_, Oct 25 2018
%Y 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 nonn,easy,fini,full
%O 0,2
%A _John Cannon_ and _N. J. A. Sloane_, Nov 30 2009
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