A266782 The growth series for the affine Coxeter (or Weyl) group [3,5] (or H_3).
1, 4, 9, 16, 25, 37, 52, 69, 88, 110, 136, 165, 196, 229, 265, 304, 345, 388, 434, 484, 537, 592, 649, 709, 772, 837, 904, 974, 1048, 1125, 1204, 1285, 1369, 1456, 1545, 1636, 1730, 1828, 1929, 2032, 2137, 2245, 2356, 2469, 2584, 2702, 2824, 2949, 3076, 3205, 3337, 3472, 3609, 3748, 3890, 4036, 4185, 4336, 4489, 4645
Offset: 0
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
- Vincenzo Librandi, Table of n, a(n) for n = 0..1000
- Index entries for linear recurrences with constant coefficients, signature (3, -3, 0, 3, -3, 0, 3, -3, 1).
Crossrefs
For the growth series for the finite group see A162495.
Programs
-
Magma
R
:=PowerSeriesRing(Integers(), 60); Coefficients(R!( (1+x)*(1+x^3)*(1+x^5)/((1-x+x^3-x^4+x^6-x^7)*(1-x)^2) )); // G. C. Greubel, Feb 04 2020 -
Maple
m:= 60; S:=series((1+x)*(1+x^3)*(1+x^5)/((1-x+x^3-x^4+x^6-x^7)*(1-x)^2), x, m+1): seq(coeff(S, x, j), j=0..m); # G. C. Greubel, Feb 04 2020
-
Mathematica
CoefficientList[Series[(1+x)*(1+x^3)*(1+x^5)/((1-x+x^3-x^4+x^6-x^7)*(1-x)^2), {x, 0, 60}], x] (* Vincenzo Librandi, Sep 07 2016 *) -
PARI
Vec( (1+x)*(1+x^3)*(1+x^5)/((1-x+x^3-x^4+x^6-x^7)*(1-x)^2) +O('x^60) ) \\ G. C. Greubel, Feb 04 2020 -
Sage
def A266782_list(prec): P.
= PowerSeriesRing(ZZ, prec) return P( (1+x)*(1+x^3)*(1+x^5)/((1-x+x^3-x^4+x^6-x^7)*(1-x)^2) ).list() A266782_list(60) # G. C. Greubel, Feb 04 2020
Formula
G.f.: (1+x)*(1+x^3)*(1+x^5)/((1-x+x^3-x^4+x^6-x^7)*(1-x)^2).