A162210 - OEIS

%I #16 Feb 21 2024 11:27:09

%S 1,7,27,77,181,371,686,1169,1862,2800,4005,5481,7210,9149,11230,13363,

%T 15442,17353,18983,20230,21013,21280,21013,20230,18983,17353,15442,

%U 13363,11230,9149,7210,5481,4005,2800,1862,1169,686,371,181,77,27,7,1,0

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

%D N. Bourbaki, Groupes et alg. de Lie, Chap. 4, 5, 6. (The group is defined in Planche IV.)

%D J. E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge, 1990. See under Poincaré polynomial.

%H <a href="/index/Gre#GROWTH">Index entries for growth series for groups</a>

%F 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 # Growth series for D_k, truncated to terms of order M. - _N. J. A. Sloane_, Aug 07 2021

%p f := proc(m::integer) (1-x^m)/(1-x) ; end proc:

%p g := proc(k,M) local a,i; global f;

%p a:=f(k)*mul(f(2*i),i=1..k-1);

%p seriestolist(series(a,x,M+1));

%p end proc;

%t n = 7;

%t x = y + y O[y]^(n^2);

%t (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 The growth series for D_k, k >= 3, are also the rows of the triangle A162206.

%Y 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 nonn

%O 0,2

%A _John Cannon_ and _N. J. A. Sloane_, Dec 01 2009