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A161717 Number of reduced words of length n in the Weyl group B_8.

%I A161717 #33 Sep 08 2022 08:45:45
%S A161717 1,8,35,112,293,664,1350,2520,4389,7216,11298,16960,24541,34376,46775,
%T A161717 62000,80241,101592,126029,153392,183373,215512,249202,283704,318171,
%U A161717 351680,383270,411984,436913,457240,472281,481520,484636,481520,472281
%N A161717 Number of reduced words of length n in the Weyl group B_8.
%C A161717 Computed with MAGMA using commands similar to those used to compute A161409.
%D A161717 J. E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge, 1990. See under Poincaré polynomial.
%D A161717 N. Bourbaki, Groupes et algèbres de Lie, Chap. 4, 5, 6. (The group is defined in Planche II.)
%H A161717 Vincenzo Librandi, <a href="/A161717/b161717.txt">Table of n, a(n) for n = 0..64</a>
%F A161717 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 A161717 seq(coeff(series(mul((1-x^(2*k))/(1-x),k=1..8),x,65), x, n), n = 0 .. 64); # _Muniru A Asiru_, Oct 25 2018
%t A161717 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)^8, {x, 0, 70}], x] (* _Vincenzo Librandi_, Aug 22 2016 *)
%o A161717 (PARI) t='t+O('t^40); Vec(prod(k=1,8,1-t^(2*k))/(1-t)^8) \\ _G. C. Greubel_, Oct 25 2018
%o A161717 (Magma) m:=40; R<t>:=PowerSeriesRing(Integers(), m); Coefficients(R!((&*[1-t^(2*k): k in [1..8]])/(1-t)^8)); // _G. C. Greubel_, Oct 25 2018
%Y A161717 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 A161717 nonn,easy,fini,full
%O A161717 0,2
%A A161717 _John Cannon_ and _N. J. A. Sloane_, Nov 30 2009