A165445 Number of reduced words of length n in Coxeter group on 27 generators S_i with relations (S_i)^2 = (S_i S_j)^9 = I.

1, 27, 702, 18252, 474552, 12338352, 320797152, 8340725952, 216858874752, 5638330743201, 146596599314100, 3811511581929675, 99099301124011500, 2576581829064137700, 66991127551503386400, 1741769316230819007600

Offset: 0

Author

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

Keywords

• nonn
• easy

Comments

The initial terms coincide with those of A170746, 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..700
• Index entries for linear recurrences with constant coefficients, signature (25,25,25,25,25,25,25,25,-325).

Programs

GAP

a:=[27, 702, 18252, 474552, 12338352, 320797152, 8340725952, 216858874752, 5638330743201];; for n in [10..20] do a[n]:=25*Sum([1..8], j-> a[n-j]) -325*a[n-9]; od; Concatenation([1], a); # G. C. Greubel, Sep 16 2019

Magma

R:=PowerSeriesRing(Integers(), 20); Coefficients(R!( (1+t)*(1-t^9)/(1-26*t+350*t^9-325*t^10) )); // G. C. Greubel, Oct 20 2018

Maple

seq(coeff(series((x^9+2*x^8+2*x^7+2*x^6+2*x^5+2*x^4+2*x^3+2*x^2+2*x+1 )/(325*x^9-25*x^8-25*x^7-25*x^6-25*x^5-25*x^4-25*x^3-25*x^2 -25*x +1),x,n+1), x, n), n = 0 .. 15); # Muniru A Asiru, Oct 21 2018

Mathematica

coxG[{9,325,-25}] (* The coxG program is at A169452 *) (* Harvey P. Dale, Sep 21 2017 *)
CoefficientList[Series[(1+t)*(1-t^9)/(1-26*t+350*t^9-325*t^10), {t, 0, 20}], t] (* G. C. Greubel, Oct 20 2018 *)

PARI

t='t+O('t^20); Vec((1+t)*(1-t^9)/(1-26*t+350*t^9-325*t^10)) \\ G. C. Greubel, Oct 20 2018

Sage

def A165445_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P((1+t)*(1-t^9)/(1-26*t+350*t^9-325*t^10)).list()
A165445_list(20) # G. C. Greubel, Sep 16 2019

Formula

G.f.: (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)/(325*t^9 - 25*t^8 - 25*t^7 - 25*t^6 - 25*t^5 - 25*t^4 - 25*t^3 - 25*t^2 - 25*t + 1).