A266759 Growth series for affine Coxeter group (or affine Weyl group) D_4.

1, 5, 14, 32, 63, 110, 179, 274, 398, 557, 754, 993, 1280, 1618, 2011, 2464, 2981, 3566, 4224, 4959, 5774, 6675, 7666, 8750, 9933, 11218, 12609, 14112, 15730, 17467, 19328, 21317, 23438, 25696, 28095, 30638, 33331, 36178, 39182, 42349, 45682, 49185, 52864, 56722, 60763, 64992, 69413, 74030, 78848, 83871, 89102, 94547, 100210, 106094

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

• M. F. Hasler, Table of n, a(n) for n = 0..499
• Index entries for linear recurrences with constant coefficients, signature (2, -1, 1, -2, 2, -2, 1, -1, 2, -1).

Crossrefs

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

Programs

PARI

A266759_vec(N=100)=Vec((1+t='t)*(1+t^3+O(t^N))*(1+t+t^2+t^3)^2/(1-t)^2/(1-t^3)/(1-t^5)) \\ M. F. Hasler, Jul 12 2018

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=4) the g.f. is (t^3+1)*(1+t)*(1+t+t^2+t^3)^2/(-1+t^5)/(-1+t)^2/(-1+t^3).