A266760 Growth series for affine Coxeter group (or affine Weyl group) D_5.

1, 6, 20, 51, 111, 216, 386, 646, 1026, 1560, 2287, 3251, 4500, 6086, 8066, 10502, 13460, 17011, 21231, 26200, 32002, 38726, 46466, 55320, 65391, 76787, 89620, 104006, 120066, 137926, 157716, 179571, 203631, 230040, 258946, 290502, 324866, 362200, 402671, 446451, 493716, 544646, 599426, 658246, 721300, 788787, 860911, 937880, 1019906

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 (4, -6, 4, -1, 0, 0, 1, -4, 6, -4, 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=5) the G.f. is -(1+t)*(t^3+1)*(1+t+t^2+t^3+t^4+t^5+t^6+t^7) / (-1+t^7) / (-1+t)^4.