A165787 Number of reduced words of length n in Coxeter group on 9 generators S_i with relations (S_i)^2 = (S_i S_j)^10 = I.

1, 9, 72, 576, 4608, 36864, 294912, 2359296, 18874368, 150994944, 1207959516, 9663675840, 77309404452, 618475217472, 4947801594624, 39582411595776, 316659283476480, 2533274193494016, 20266192953409536, 162129538870935552

Offset: 0

Author

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

Keywords

• nonn

Comments

The initial terms coincide with those of A003951, 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 (7,7,7,7,7,7,7,7,7,-28).

Programs

GAP

a:=[9, 72, 576, 4608, 36864, 294912, 2359296, 18874368, 150994944, 1207959516];; for n in [11..20] do a[n]:=7*Sum([1..9], j-> a[n-j]) -28*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)/(1-8*t+35*t^10-28*t^11) )); // G. C. Greubel, Sep 22 2019

Maple

seq(coeff(series((1+t)*(1-t^10)/(1-8*t+35*t^10-28*t^11), t, n+1), t, n), n = 0..20); # G. C. Greubel, Aug 10 2019

Mathematica

CoefficientList[Series[(1+t)*(1-t^10)/(1-8*t+35*t^10-28*t^11), {t, 0, 20}], t] (* G. C. Greubel, Apr 08 2016 *)
coxG[{10, 28, -7}] (* 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)/(1-8*t+35*t^10-28*t^11)) \\ G. C. Greubel, Sep 22 2019

Sage

def A165787_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P((1+t)*(1-t^10)/(1-8*t+35*t^10-28*t^11)).list()
A165787_list(20) # 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)/(28*t^10 - 7*t^9 - 7*t^8 - 7*t^7 - 7*t^6 - 7*t^5 - 7*t^4 - 7*t^3 - 7*t^2 - 7*t + 1).