A164685 Number of reduced words of length n in Coxeter group on 41 generators S_i with relations (S_i)^2 = (S_i S_j)^7 = I.

1, 41, 1640, 65600, 2624000, 104960000, 4198400000, 167935999180, 6717439934400, 268697596064820, 10747903790145600, 429916149507936000, 17196645896401920000, 687865832499456000000, 27514633165713408671580

Offset: 0

Author

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

Keywords

• nonn

Comments

The initial terms coincide with those of A170760, 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..620
• Index entries for linear recurrences with constant coefficients, signature (39,39,39,39,39,39,-780).

Programs

GAP

a:=[41,1640,65600,2624000,104960000,4198400000,167935999180];; for n in [8..20] do a[n]:=39*(a[n-1] +a[n-2]+a[n-3]+a[n-4]+a[n-5]+a[n-6]) -780*a[n-7]; od; Concatenation([1], a); # G. C. Greubel, Sep 15 2019

Magma

R:=PowerSeriesRing(Integers(), 20); Coefficients(R!( (1+t)*(1-t^7)/(1-40*t+819*t^7-780*t^8) )); // G. C. Greubel, Sep 15 2019

Maple

seq(coeff(series((1+t)*(1-t^7)/(1-40*t+819*t^7-780*t^8), t, n+1), t, n), n = 0..20); # G. C. Greubel, Sep 15 2019

Mathematica

CoefficientList[Series[(1+t)*(1-t^7)/(1-40*t+819*t^7-780*t^8), {t, 0, 20}], t] (* G. C. Greubel, Sep 15 2019 *)
coxG[{7, 780, -39}] (* The coxG program is at A169452 *) (* G. C. Greubel, Sep 15 2019 *)

PARI

my(t='t+O('t^20)); Vec((1+t)*(1-t^7)/(1-40*t+819*t^7-780*t^8)) \\ G. C. Greubel, Sep 15 2019

Sage

def A164685_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P((1+t)*(1-t^7)/(1-40*t+819*t^7-780*t^8)).list()
A164685_list(20) # G. C. Greubel, Sep 15 2019

Formula

G.f.: (t^7 + 2*t^6 + 2*t^5 + 2*t^4 + 2*t^3 + 2*t^2 + 2*t + 1)/(780*t^7 - 39*t^6 - 39*t^5 - 39*t^4 - 39*t^3 - 39*t^2 - 39*t + 1).