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A162208 Number of reduced words of length n in the Weyl group D_5.

%I A162208 #29 Mar 21 2025 18:01:48
%S A162208 1,5,14,30,54,85,120,155,185,205,212,205,185,155,120,85,54,30,14,5,1
%N A162208 Number of reduced words of length n in the Weyl group D_5.
%D A162208 N. Bourbaki, Groupes et algèbres de Lie, Chap. 4, 5, 6. (The group is defined in Planche IV.)
%D A162208 J. E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge, 1990. See under Poincaré polynomial.
%H A162208 <a href="/index/Gre#GROWTH">Index entries for growth series for groups</a>
%F A162208 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 A162208 A162208g := proc(m::integer)
%p A162208     (1-x^m)/(1-x) ;
%p A162208 end proc:
%p A162208 A162208 := proc(n,k)
%p A162208     g := A162208g(k);
%p A162208     for m from 2 to 2*k-2 by 2 do
%p A162208         g := g*A162208g(m) ;
%p A162208     end do:
%p A162208     g := expand(g) ;
%p A162208     coeftayl(g,x=0,n) ;
%p A162208 end proc:
%p A162208 seq( A162208(n,5),n=0..60) ; # _R. J. Mathar_, Jan 19 2016
%t A162208 n = 5;
%t A162208 x = y + y O[y]^(n^2);
%t A162208 (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 A162208 Row 5 of A162206.
%Y A162208 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 A162208 nonn,fini,full
%O A162208 0,2
%A A162208 _John Cannon_ and _N. J. A. Sloane_, Dec 01 2009