A162378 - OEIS

%I #13 Feb 21 2024 11:37:49

%S 1,31,495,5425,45879,319145,1900920,9965384,46909324,201295028,

%T 796809245,2937251395,10161553364,33205476524,103050077489,

%U 305131440111,865481871426,2359754902590,6203436293890,15765840836350,38828731002622

%N Number of reduced words of length n in the Weyl group D_31.

%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)*Product_{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 f[m_] := (1-x^m)/(1-x);

%t With[{k = 31}, CoefficientList[f[k]*Product[f[2i], {i, 1, k-1}] + O[x]^(k-10), x]] (* _Jean-François Alcover_, Feb 15 2023, after Maple code *)

%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