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Growth series for affine Coxeter group (or affine Weyl group) D_7.
1

%I #14 Feb 18 2024 12:37:07

%S 1,8,35,113,301,700,1472,2864,5236,9094,15128,24255,37669,56896,83853,

%T 120913,170975,237539,324787,437668,581987,764501,993020,1276513,

%U 1625220,2050768,2566292,3186562,3928115,4809392,5850881,7075264,8507569,10175328,12108740,14340839,16907667,19848452,23205791,27025840,31358509,36257661

%N Growth series for affine Coxeter group (or affine Weyl group) D_7.

%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="/A266762/b266762.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_25">Index entries for linear recurrences with constant coefficients</a>, signature (5, -10, 10, -5, 1, 0, 0, 0, 1, -5, 11, -15, 15, -11, 5, -1, 0, 0, 0, -1, 5, -10, 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=7) the G.f. is -(1+t+t^2+t^3)*(1+t)*(1+t+t^2+t^3+t^4+t^5+t^6+t^7)*(t^5+1)*(t^9+t^6+t^3+1)/(-1+t^11)/(-1+t^9)/(-1+t)^5.

%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