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%I #23 Feb 21 2024 11:47:40
%S 1,41,860,12300,134889,1209377,9230207,61657399,367846424,1990342376,
%T 9885562358,45508669878,195729780567,791712506207,3028721321382,
%U 11010682764150,38197208930405,126905454993645,405078061871575
%N Number of reduced words of length n in the Weyl group D_41.
%C Computed with MAGMA using commands similar to those used to compute A161409.
%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 Robert Israel, <a href="/A162403/b162403.txt">Table of n, a(n) for n = 0..1640</a>
%H <a href="/index/Gre#GROWTH">Index entries for growth series for groups</a>
%F G.f. for D_m is the polynomial f(n) * Product( f(2i), i=1..n-1 )/ f(1)^n, where f(k) = 1-x^k. Only finitely many terms are nonzero. This is a row of the triangle in A162206.
%p f:= k -> 1-x^k:
%p g:= n -> f(n)*mul(f(2*i),i=1..n-1)/f(1)^n:
%p S:= expand(normal(g(41))):
%p seq(coeff(S,x,j),j=0..degree(S,x)); # _Robert Israel_, Oct 07 2015
%t n = 41;
%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 Cf. A161409.
%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,fini,full
%O 0,2
%A _John Cannon_ and _N. J. A. Sloane_, Dec 01 2009
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