A266763 Growth series for affine Coxeter group (or affine Weyl group) D_8.

1, 9, 44, 157, 458, 1158, 2629, 5487, 10703, 19746, 34764, 58808, 96104, 152379, 235247, 354661, 523436, 757850, 1078327, 1510207, 2084608, 2839386, 3820199, 5081680, 6688726, 8717906, 11258994, 14416631, 18312124, 23085388, 28897036, 35930623, 44395047, 54527114, 66594270, 80897509, 97774461, 117602666, 140803039, 167843531

Offset: 0

Author

N. J. A. Sloane, Jan 10 2016

Keywords

• nonn

References

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

Links

• Ray Chandler, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, 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).

Crossrefs

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

Formula

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].

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).