%I #15 Feb 18 2024 12:38:59

%S 1,9,44,157,458,1158,2629,5487,10703,19746,34764,58808,96104,152379,

%T 235247,354661,523436,757850,1078327,1510207,2084608,2839386,3820199,

%U 5081680,6688726,8717906,11258994,14416631,18312124,23085388,28897036,35930623,44395047,54527114,66594270,80897509,97774461,117602666,140803039,167843531

%N Growth series for affine Coxeter group (or affine Weyl group) D_8.

%D N. Bourbaki, Groupes et Algèbres de Lie, Chap. 4, 5 and 6, Hermann, Paris, 1968. See Chap. VI, Section 4, Problem 10b, page 231, W_a(t).

%D J. E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge, 1990. See Table 3.1, page 59.

%H Ray Chandler, <a href="/A266763/b266763.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_42">Index entries for linear recurrences with constant coefficients</a>, signature (5, -10, 9, 0, -9, 9, 1, -14, 20, -14, 2, 4, 2, -14, 19, -10, -4, 7, 5, -21, 28, -21, 5, 7, -4, -10, 19, -14, 2, 4, 2, -14, 20, -14, 1, 9, -9, 0, 9, -10, 5, -1).

%F The growth series for the affine Coxeter group of type D_k (k >= 3) has g.f. = Product_i (1-x^{m_i+1})/((1-x)*(1-x^{m_i})) where the m_i are [1,3,5,...,2k-3,k-1].

%F Here (k=8) the G.f. is (1+t+t^2+t^3+t^4+t^5+t^6+t^7)^2*(1+t)*(1+t+t^2+t^3)*(t^3+1)*(t^5+1)*(t^9+t^6+t^3+1)*(t^7+1)/(-1+t^13)/(-1+t^11)/(t^7-t^6+t^4-t^3+t-1)/(-1+t)^4/(-1+t^7).

%Y The growth series for the affine Coxeter groups D_3 through D_12 are A005893 and A266759-A266767.

%K nonn

%O 0,2

%A _N. J. A. Sloane_, Jan 10 2016