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A162212 Number of reduced words of length n in the Weyl group D_9.

%I A162212 #25 Mar 17 2025 15:56:22
%S A162212 1,9,44,156,449,1113,2463,4983,9372,16587,27877,44802,69231,103314,
%T A162212 149425,210075,287796,384999,503812,645906,812319,1003290,1218116,
%U A162212 1455045,1711216,1982655,2264333,2550288,2833809,3107676,3364445,3596763,3797695,3961044,4081645
%N A162212 Number of reduced words of length n in the Weyl group D_9.
%D A162212 N. Bourbaki, Groupes et alg. de Lie, Chap. 4, 5, 6. (The group is defined in Planche IV.)
%D A162212 J. E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge, 1990. See under Poincaré polynomial.
%H A162212 Andrew Howroyd, <a href="/A162212/b162212.txt">Table of n, a(n) for n = 0..72</a>
%H A162212 <a href="/index/Gre#GROWTH">Index entries for growth series for groups</a>
%F A162212 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 A162212 A162212g := proc(m::integer)
%p A162212     (1-x^m)/(1-x) ;
%p A162212 end proc:
%p A162212 A162212 := proc(n,k)
%p A162212     g := A162212g(k);
%p A162212     for m from 2 to 2*k-2 by 2 do
%p A162212         g := g*A162212g(m) ;
%p A162212     end do:
%p A162212     g := expand(g) ;
%p A162212     coeftayl(g,x=0,n) ;
%p A162212 end proc:
%p A162212 seq( A162212(n,9),n=0..30) ; # _R. J. Mathar_, Jan 19 2016
%p A162212 # Growth series for D_k, truncated to terms of order M. - _N. J. A. Sloane_, Aug 07 2021
%p A162212 f := proc(m::integer) (1-x^m)/(1-x) ; end proc:
%p A162212 g := proc(k,M) local a,i; global f;
%p A162212 a:=f(k)*mul(f(2*i),i=1..k-1);
%p A162212 seriestolist(series(a,x,M+1));
%p A162212 end proc;
%t A162212 n = 9;
%t A162212 x = y + y O[y]^(n^2);
%t A162212 (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 A162212 Row 9 of A162206.
%Y A162212 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 A162212 nonn,fini,full
%O A162212 0,2
%A A162212 _John Cannon_ and _N. J. A. Sloane_, Dec 01 2009