A267176 The growth series for the affine Weyl group E_8.

1, 9, 44, 156, 450, 1122, 2508, 5149, 9875, 17910, 31000, 51567, 82892, 129330, 196561, 291880, 424528, 606067, 850803, 1176260, 1603708, 2158748, 2871957, 3779597, 4924393, 6356383, 8133842, 10324283, 13005538, 16266923, 20210492, 24952383, 30624256, 37374826, 45371496, 54802094, 65876718, 78829693, 93921640, 111441659, 131709633

Offset: 0

Author

N. J. A. Sloane, Jan 11 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).
• H. S. M. Coxeter and W. O. J. Moser, Generators and Relations for Discrete Groups, 4th ed., Springer-Verlag, NY, reprinted 1984, Table 10.
• J. E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge, 1990. See under Poincaré polynomial; also 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 (2, -1, 0, 0, 0, 0, 0, 0, 0, 0, 1, -2, 2, -2, 1, 0, 1, -2, 2, -2, 1, 0, 1, -3, 3, -1, 0, -1, 3, -5, 5, -3, 2, -2, 2, -3, 4, -2, 0, -2, 5, -6, 6, -4, 1, -1, 4, -6, 5, -3, 2, -3, 6, -7, 4, -2, 2, -2, 4, -7, 6, -3, 2, -3, 5, -6, 4, -1, 1, -4, 6, -6, 5, -2, 0, -2, 4, -3, 2, -2, 2, -3, 5, -5, 3, -1, 0, -1, 3, -3, 1, 0, 1, -2, 2, -2, 1, 0, 1, -2, 2, -2, 1, 0, 0, 0, 0, 0, 0, 0, 0, -1, 2, -1).

Crossrefs

For the growth series for the finite group see A162494.

Formula

G.f. = t1/t2, where t1 is

(1+t)*(1+t+t^2+t^3+t^4+t^5+t^6+t^7)

*(1+t+t^2+t^3+t^4+t^5+t^6+t^7+t^8+t^9+t^10+t^11)

*(1+t+t^2+t^3+t^4+t^5+t^6+t^7+t^8+t^9+t^10+t^11+t^12+t^13)

*(1+t+t^2+t^3+t^4+t^5+t^6+t^7+t^8+t^9+t^10+t^11+t^12+t^13+t^14+t^15+t^16+t^17)

*(1+t+t^2+t^3+t^4+t^5+t^6+t^7+t^8+t^9+t^10+t^11+t^12+t^13+t^14+t^15+t^16+t^17+t^18+t^19)

*(1+t+t^2+t^3+t^4+t^5+t^6+t^7+t^8+t^9+t^10+t^11+t^12+t^13+t^14+t^15+t^16+t^17+t^18+t^19+t^20+t^21+t^22+t^23)

*(1+t^20+t^21+t^22+t^23+t^18+t^19+t^2+t^3+t^4+t^5+t^6+t^12+t^13+t^7+t^8+t^9+t^10+t^11+t^14+t^15+t^16+t^17+t+t^24+t^25+t^26+t^27+t^28+t^29),

and t2 = (1-t)*(1-t^7)*(1-t^11)*(1-t^13)*(1-t^17)*(1-t^19)*(1-t^23)*(1-t^29).