A266784 The growth series for the affine Weyl group F_4.

1, 5, 14, 30, 55, 92, 144, 214, 305, 419, 559, 729, 933, 1173, 1451, 1771, 2137, 2552, 3018, 3537, 4111, 4744, 5441, 6205, 7037, 7940, 8919, 9977, 11116, 12338, 13646, 15043, 16533, 18120, 19805, 21590, 23480, 25480, 27592, 29817, 32158, 34618, 37200, 39908, 42745, 45713, 48815, 52056, 55439, 58965, 62637, 66459, 70434, 74564, 78852, 83301

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

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (1, 0, 0, 0, 1, -1, 1, -1, 0, 0, 1, -2, 1, 0, 0, -1, 1, -1, 1, 0, 0, 0, 1, -1).

Crossrefs

For the growth series for the finite group see A162496.

Programs

Magma

R:=PowerSeriesRing(Integers(), 60); Coefficients(R!( (1-x^2)*(1-x^6)*(1-x^8)*(1-x^12)/((1-x)^5*(1-x^5)*(1-x^7)*(1-x^11)) )); // G. C. Greubel, Feb 04 2020

Maple

m:=30; S:=series((1-x^2)*(1-x^6)*(1-x^8)*(1-x^12)/((1-x)^5*(1-x^5)*(1-x^7)*(1-x^11)), x, m+1): seq(coeff(S, x, j), j=0..m); # G. C. Greubel, Feb 04 2020

Mathematica

CoefficientList[Series[(1-x^2)*(1-x^6)*(1-x^8)*(1-x^12)/((1-x)^5*(1-x^5)*(1-x^7)*(1-x^11)), {x,0,60}], x] (* G. C. Greubel, Feb 04 2020 *)

PARI

Vec( (1-x^2)*(1-x^6)*(1-x^8)*(1-x^12)/((1-x)^5*(1-x^5)*(1-x^7)*(1-x^11)) +O('x^60) ) \\ G. C. Greubel, Feb 04 2020

Sage

def A077952_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1-x^2)*(1-x^6)*(1-x^8)*(1-x^12)/((1-x)^5*(1-x^5)*(1-x^7)*(1-x^11)) ).list()
A077952_list(60) # G. C. Greubel, Feb 04 2020

Formula

G.f.: (1+t)*(1+t+t^2+t^3+t^4+t^5)*(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)*(1-t^5)*(1-t^7)*(1-t^11)).

G.f.: (1-x^2)*(1-x^6)*(1-x^8)*(1-x^12)/((1-x)^5*(1-x^5)*(1-x^7)*(1-x^11)). - G. C. Greubel, Feb 04 2020