A165757 Number of reduced words of length n in Coxeter group on 5 generators S_i with relations (S_i)^2 = (S_i S_j)^10 = I.

1, 5, 20, 80, 320, 1280, 5120, 20480, 81920, 327680, 1310710, 5242800, 20971050, 83883600, 335532000, 1342118400, 5368435200, 21473587200, 85893734400, 343572480000, 1374280089690, 5497081037820, 21988166868630, 87952038348420

Offset: 0

Author

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

Keywords

• nonn

Comments

The initial terms coincide with those of A003947, 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 (3,3,3,3,3,3,3,3,3,-6).

Programs

GAP

a:=[5, 20, 80, 320, 1280, 5120, 20480, 81920, 327680, 1310710];; for n in [11..30] do a[n]:=3*Sum([1..9], j-> a[n-j]) -6*a[n-10]; od; Concatenation([1], a); # G. C. Greubel, Sep 17 2019

Magma

R:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1+t)*(1-t^10)/(1-4*t+9*t^10-6*t^11) )); // G. C. Greubel, Sep 17 2019

Maple

seq(coeff(series((1+t)*(1-t^10)/(1-4*t+9*t^10-6*t^11), t, n+1), t, n), n = 0..30); # G. C. Greubel, Sep 17 2019

Mathematica

CoefficientList[Series[(1+t)*(1-t^10)/(1-4*t+9*t^10-6*t^11), {t, 0, 30}], t] (* G. C. Greubel, Apr 07 2016 *)
coxG[{10, 6, -3}] (* The coxG program is at A169452 *) (* G. C. Greubel, Sep 17 2019 *)

PARI

my(t='t+O('t^30)); Vec((1+t)*(1-t^10)/(1-4*t+9*t^10-6*t^11)) \\ G. C. Greubel, Sep 17 2019

Sage

def A165757_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P((1+t)*(1-t^10)/(1-4*t+9*t^10-6*t^11)).list()
A165757_list(30) # G. C. Greubel, Sep 17 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)/(6*t^10 - 3*t^9 - 3*t^8 - 3*t^7 - 3*t^6 - 3*t^5 - 3*t^4 - 3*t^3 - 3*t^2 - 3*t + 1).