A266761 Growth series for affine Coxeter group (or affine Weyl group) D_6.

1, 7, 27, 78, 188, 400, 777, 1406, 2403, 3917, 6136, 9293, 13670, 19603, 27485, 37773, 50993, 67744, 88703, 114628, 146366, 184857, 231139, 286352, 351742, 428669, 518610, 623164, 744055, 883138, 1042406, 1223994, 1430184, 1663408, 1926254, 2221471, 2551974, 2920848, 3331353, 3786930, 4291206, 4847999, 5461321, 6135384, 6874604

Offset

0,2

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, 3, 3, -5, -1, 10, -13, 7, 2, -6, 2, 7, -13, 10, -1, -5, 3, 3, -6, 4, -1).

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

Crossrefs

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

Sequence in context: A005585 A161410 A267169 * A027180 A036597 A338230

Adjacent sequences: A266758 A266759 A266760 * A266762 A266763 A266764

Keyword

nonn

Author

N. J. A. Sloane, Jan 10 2016

Status

approved