A266785 The growth series for the affine Weyl group E_7.

1, 8, 35, 112, 294, 673, 1393, 2668, 4803, 8218, 13476, 21315, 32684, 48782, 71101, 101473, 142121, 195714, 265426, 354999, 468809, 611936, 790238, 1010430, 1280166, 1608124, 2004094, 2479071, 3045353, 3716642, 4508148, 5436696, 6520838, 7780968, 9239441, 10920695, 12851378, 15060479, 17579463, 20442410, 23686158, 27350450, 31478083

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 (4, -6, 4, -1, 0, 0, 0, 0, 0, 0, 1, -4, 7, -8, 7, -4, 2, -4, 6, -4, 1, 0, 0, -1, 4, -6, 4, -2, 4, -7, 8, -7, 4, -1, 0, 0, 0, 0, 0, 0, 1, -4, 6, -4, 1).

Crossrefs

For the growth series for the finite group see A162493.

Programs

Magma

R:=PowerSeriesRing(Integers(), 45); Coefficients(R!( ((1-x^2)*(1-x^18)/((1-x)^8*(1-x^17)))*(&*[(1-x^(2*j+6))/(1-x^(2*j+5)): j in [0..4]]) )); // G. C. Greubel, Feb 04 2020

Maple

m:=45; S:=series(((1-x^2)*(1-x^18)/((1-x)^8*(1-x^17)))*mul((1-x^(2*j+6))/(1-x^(2*j+5)), j=0..4)), 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^18)/((1-x)^8*(1-x^17)))*Product[(1-x^(2*j + 6))/(1-x^(2*j+5)), {j,0,4}], {x,0,45}], x] (* G. C. Greubel, Feb 04 2020 *)

PARI

Vec( ((1-x^2)*(1-x^18)/((1-x)^8*(1-x^17)))*prod(j=0,4, (1-x^(2*j+6))/(1-x^(2*j+5))) +O('x^45) ) \\ G. C. Greubel, Feb 04 2020

Sage

def A266785_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( ((1-x^2)*(1-x^18)/((1-x)^8*(1-x^17)))*product((1-x^(2*j+6))/(1-x^(2*j+5)) for j in (0..4)) ).list()
A266785_list(45) # G. C. Greubel, Feb 04 2020

Formula

G.f.: (1 +t)*(1 +t^3)*(1 +t^5)*(1 +t^7)*(1 +t +t^2 +t^3 +t^4 +t^5 +t^6 +t^7)*(1 +t +t^2 +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)/((1-t)^4*(1-t^11)*(1-t^13)*(1-t^17)).

G.f.: ((1-x^2)*(1-x^18)/((1-x)^8*(1-x^17)))*Product_{j=0..4} (1-x^(2*j+6))/(1-x^(2*j+5)). - G. C. Greubel, Feb 05 2020