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A162297 Number of reduced words of length n in the Weyl group D_12.

%I A162297 #29 Mar 17 2025 17:05:45
%S A162297 1,12,77,352,1286,3992,10933,27092,61841,131768,264759,505660,923857,
%T A162297 1623104,2753895,4528612,7239585,11280072,17168009,25572196,37340381,
%U A162297 53528488,75430016,104604424,142903123,192491532,255865533,335860592,435651810,558743240,708944960
%N A162297 Number of reduced words of length n in the Weyl group D_12.
%D A162297 N. Bourbaki, Groupes et Algèbres de Lie, Chap. 4, 5 and 6, Hermann, Paris, 1968. See Chap. VI, Section 4, Problem 10a, page 231, W(t).
%D A162297 J. E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge, 1990. See under Poincaré polynomial.
%H A162297 Andrew Howroyd, <a href="/A162297/b162297.txt">Table of n, a(n) for n = 0..132</a>
%H A162297 <a href="/index/Gre#GROWTH">Index entries for growth series for groups</a>
%F A162297 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 A162297 # Growth series for D_k, truncated to terms of order M. - _N. J. A. Sloane_, Aug 07 2021
%p A162297 f := proc(m::integer) (1-x^m)/(1-x) ; end proc:
%p A162297 g := proc(k,M) local a,i; global f;
%p A162297 a:=f(k)*mul(f(2*i),i=1..k-1);
%p A162297 seriestolist(series(a,x,M+1));
%p A162297 end proc;
%t A162297 n = 12;
%t A162297 x = y + y O[y]^(n^2);
%t A162297 (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 A162297 Row 12 of A162206.
%Y A162297 Growth series for groups D_n, n = 3,...,50: A161435, A162207, A162208, A162209, A162210, A162211, A162212, A162248, A162288, 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 A162297 nonn,fini,full
%O A162297 0,2
%A A162297 _John Cannon_ and _N. J. A. Sloane_, Dec 01 2009
%E A162297 Entry revised by _N. J. A. Sloane_, Jan 17 2016