A165786 Number of reduced words of length n in Coxeter group on 8 generators S_i with relations (S_i)^2 = (S_i S_j)^10 = I.

1, 8, 56, 392, 2744, 19208, 134456, 941192, 6588344, 46118408, 322828828, 2259801600, 15818609856, 110730259584, 775111751232, 5425781797632, 37980469356480, 265863262906752, 1861042682227008, 13027297668747264

Offset: 0

Author

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

Keywords

• nonn

Comments

The initial terms coincide with those of A003950, 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 (6,6,6,6,6,6,6,6,6,-21).

Programs

GAP

a:=[8, 56, 392, 2744, 19208, 134456, 941192, 6588344, 46118408, 322828828];; for n in [11..20] do a[n]:=6*Sum([1..9], j-> a[n-j]) -21*a[n-10]; od; Concatenation([1], a); # G. C. Greubel, Sep 22 2019

Magma

R:=PowerSeriesRing(Integers(), 20); Coefficients(R!( (1+t)*(1-t^10)/(1-7*t+27*t^10-21*t^11) )); // G. C. Greubel, Sep 22 2019

Maple

seq(coeff(series((1+t)*(1-t^10)/(1-7*t+27*t^10-21*t^11), t, n+1), t, n), n = 0..20); # G. C. Greubel, Sep 22 2019

Mathematica

With[{num=Total[2t^Range[9]]+t^10+1,den=Total[-6 t^Range[9]]+21t^10+1}, CoefficientList[Series[num/den,{t,0,30}],t]] (* Harvey P. Dale, Oct 20 2011 *)
CoefficientList[Series[(1+t)*(1-t^10)/(1-7*t+27*t^10-21*t^11), {t,0,20}], t] (* or *) coxG[{10, 21, -6}] (* The coxG program is at A169452 *) (* G. C. Greubel, Sep 22 2019 *)

PARI

my(t='t+O('t^20)); Vec((1+t)*(1-t^10)/(1-7*t+27*t^10-21*t^11)) \\ G. C. Greubel, Sep 22 2019

Sage

def A165786_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1+t)*(1-t^10)/(1-7*t+27*t^10-21*t^11) ).list()
A165786_list(30) # G. C. Greubel, Sep 22 2019

Formula

G.f.: (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)/(21*t^10 - 6*t^9 - 6*t^8 - 6*t^7 - 6*t^6 - 6*t^5 - 6*t^4 - 6*t^3 - 6*t^2 - 6*t + 1).