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A161435 Number of reduced words of length n in the Weyl group A_3 (or D_3).

%I A161435 #33 Feb 21 2024 11:24:22
%S A161435 1,3,5,6,5,3,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
%T A161435 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 A161435 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 A161435 Number of reduced words of length n in the Weyl group A_3 (or D_3).
%C A161435 a(n) is also the number of vertices of a truncated octahedron (the Voronoi cell for the lattice A_3*) at edge distance n from a given vertex. See also row 4 of the triangle in A008302. - _N. J. A. Sloane_, Oct 12 2015, corrected Aug 26 2016.
%C A161435 If the zeros are omitted, this is the coordination sequence for the truncated octahedron (see Karzes link). - _N. J. A. Sloane_, Jan 08 2020
%C A161435 Computed with Magma using commands similar to those used to compute A161409.
%D A161435 N. Bourbaki, Groupes et alg. de Lie, Chap. 4, 5, 6. (The group is defined in Planche I.)
%D A161435 J. E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge, 1990. See under Poincaré polynomial.
%H A161435 Tom Karzes, <a href="/A298808/a298808.html">Polyhedron Coordination Sequences</a>
%H A161435 <a href="/index/Gre#GROWTH">Index entries for growth series for groups</a>
%F A161435 G.f. for A_m is the polynomial Product_{k=1..m} (1-x^(k+1))/(1-x). Only finitely many terms are nonzero. This is a row of the triangle in A008302.
%p A161435 # Growth series for D_k, truncated to terms of order M. - _N. J. A. Sloane_, Aug 07 2021
%p A161435 f := proc(m::integer) (1-x^m)/(1-x) ; end proc:
%p A161435 g := proc(k,M) local a,i; global f;
%p A161435 a:=f(k)*mul(f(2*i),i=1..k-1);
%p A161435 seriestolist(series(a,x,M+1));
%p A161435 end proc;
%t A161435 CoefficientList[Series[(1 - x^2) (1 - x^3) (1 - x^4) / (1 - x)^3, {x, 0, 20}], x] (* _Vincenzo Librandi_, Aug 23 2016 *)
%Y A161435 Cf. A008302, A161409.
%Y A161435 Growth series for groups D_n, n = 3,...,50: A161435, A162207, A162208, A162209, A162210, A162211, A162212, A162248, A162288, A162297, A162300, A162301, A162321, A162327, A162328, A162346, A162347, A162359, A162360, A162364, A162365, A162366, A162367, A162368, A162369, A162370, A162376, A162377, A162378, A162379, A162380, A162381, A162384, A162388, A162389, A162392, A162399, A162402, A162403, A162411, A162412, A162413, A162418, A162452, A162456, A162461, A162469, A162492; also A162206.
%K A161435 nonn,easy
%O A161435 0,2
%A A161435 _John Cannon_ and _N. J. A. Sloane_, Nov 30 2009