A166431 Number of reduced words of length n in Coxeter group on 37 generators S_i with relations (S_i)^2 = (S_i S_j)^11 = I.

1, 37, 1332, 47952, 1726272, 62145792, 2237248512, 80540946432, 2899474071552, 104381066575872, 3757718396731392, 135277862282329446, 4870003042163836080, 175320109517897236410, 6311523942644269461840

Offset: 0

Author

John Cannon and N. J. A. Sloane, Dec 03 2009

Keywords

• nonn

Comments

The initial terms coincide with those of A170756, although the two sequences are eventually different.

Computed with MAGMA using commands similar to those used to compute A154638.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..500
• Index entries for linear recurrences with constant coefficients, signature (35,35,35,35,35,35,35,35,35,35,-630).

Crossrefs

Cf. A154638, A169452, A170756.

Programs

Magma

R:=PowerSeriesRing(Integers(), 30);
Coefficients(R!( (1+x)*(1-x^11)/(1-36*x+665*x^11-630*x^12) )); // G. C. Greubel, Jul 25 2024

Mathematica

With[{p=630, q=35}, CoefficientList[Series[(1+t)*(1-t^11)/(1-(q+1)*t + (p+q)*t^11-p*t^12), {t,0,40}], t]] (* G. C. Greubel, May 14 2016; Jul 25 2024 *)
coxG[{11, 630, -35. 30}] (* The coxG program is at A169452 *) (* G. C. Greubel, Jul 25 2024 *)

SageMath

def A166431_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1+x)*(1-x^11)/(1-36*x+665*x^11-630*x^12) ).list()
A166431_list(30) # G. C. Greubel, Jul 25 2024

Formula

G.f.: (t^11 + 2*t^10 + 2*t^9 + 2*t^8 + 2*t^7 + 2*t^6 + 2*t^5 + 2*t^4 + 2*t^3 + 2*t^2 + 2*t + 1)/(630*t^11 - 35*t^10 - 35*t^9 - 35*t^8 - 35*t^7 - 35*t^6 - 35*t^5 - 35*t^4 - 35*t^3 - 35*t^2 - 35*t + 1).

From G. C. Greubel, Jul 25 2024: (Start)

a(n) = 35*Sum_{j=1..10} a(n-j) - 630*a(n-11).

G.f.: (1+x)*(1-x^11)/(1 - 36*x + 665*x^11 - 630*x^12). (End)