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A162210 Number of reduced words of length n in the Weyl group D_7.

%I A162210 #21 Mar 21 2025 17:09:09
%S A162210 1,7,27,77,181,371,686,1169,1862,2800,4005,5481,7210,9149,11230,13363,
%T A162210 15442,17353,18983,20230,21013,21280,21013,20230,18983,17353,15442,
%U A162210 13363,11230,9149,7210,5481,4005,2800,1862,1169,686,371,181,77,27,7,1
%N A162210 Number of reduced words of length n in the Weyl group D_7.
%D A162210 N. Bourbaki, Groupes et alg. de Lie, Chap. 4, 5, 6. (The group is defined in Planche IV.)
%D A162210 J. E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge, 1990. See under Poincaré polynomial.
%H A162210 <a href="/index/Gre#GROWTH">Index entries for growth series for groups</a>
%F A162210 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 A162210 # Growth series for D_k, truncated to terms of order M. - _N. J. A. Sloane_, Aug 07 2021
%p A162210 f := proc(m::integer) (1-x^m)/(1-x) ; end proc:
%p A162210 g := proc(k,M) local a,i; global f;
%p A162210 a:=f(k)*mul(f(2*i),i=1..k-1);
%p A162210 seriestolist(series(a,x,M+1));
%p A162210 end proc;
%t A162210 n = 7;
%t A162210 x = y + y O[y]^(n^2);
%t A162210 (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 A162210 Row 7 of A162206.
%Y A162210 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.
%K A162210 nonn,fini,full
%O A162210 0,2
%A A162210 _John Cannon_ and _N. J. A. Sloane_, Dec 01 2009