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A162207 Number of reduced words of length n in the Weyl group D_4.

%I A162207 #18 Feb 21 2024 11:25:17
%S A162207 1,4,9,16,23,28,30,28,23,16,9,4,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
%T A162207 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 A162207 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 A162207 Number of reduced words of length n in the Weyl group D_4.
%D A162207 N. Bourbaki, Groupes et alg. de Lie, Chap. 4, 5, 6. (The group is defined in Planche IV.)
%D A162207 J. E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge, 1990. See under Poincaré polynomial.
%H A162207 <a href="/index/Gre#GROWTH">Index entries for growth series for groups</a>
%F A162207 The growth series for D_k is the polynomial f(k)*Prod_{i=1..k-1} f(2*i), where f(m) = (1-x^m)/(1-x) [Corrected by _N. J. A. Sloane_, Aug 07 2021]. This is a row of the triangle in A162206.
%p A162207 # Growth series for D_k, truncated to terms of order M. - _N. J. A. Sloane_, Aug 07 2021
%p A162207 f := proc(m::integer) (1-x^m)/(1-x) ; end proc:
%p A162207 g := proc(k,M) local a,i; global f;
%p A162207 a:=f(k)*mul(f(2*i),i=1..k-1);
%p A162207 seriestolist(series(a,x,M+1));
%p A162207 end proc;
%t A162207 n = 4;
%t A162207 x = y + y O[y]^(n^2);
%t A162207 (1-x^n) Product[1-x^(2k), {k, 1, n-1}]/(1-x)^n // CoefficientList[#, y]& (* _Jean-François Alcover_, Mar 25 2020, from A162206 *)
%Y A162207 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 A162207 nonn
%O A162207 0,2
%A A162207 _John Cannon_ and _N. J. A. Sloane_, Dec 01 2009