A165266 Number of reduced words of length n in Coxeter group on 12 generators S_i with relations (S_i)^2 = (S_i S_j)^9 = I.

1, 12, 132, 1452, 15972, 175692, 1932612, 21258732, 233846052, 2572306506, 28295370840, 311249071320, 3423739697400, 37661135713080, 414272482302360, 4556997189369240, 50126967807537720, 551396631852151800

Offset: 0

Author

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

Keywords

• nonn

Comments

The initial terms coincide with those of A003954, 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..955
• Index entries for linear recurrences with constant coefficients, signature (10,10,10,10,10,10,10,10,-55).

Programs

GAP

a:=[12, 132, 1452, 15972, 175692, 1932612, 21258732, 233846052, 2572306506];; for n in [10..30] do a[n]:=10*Sum([1..8], j-> a[n-j]) -55*a[n-9]; od; Concatenation([1], a); # G. C. Greubel, Sep 25 2019

Magma

R:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1+t)*(1-t^9)/(1-11*t+65*t^9-55*t^10) )); // G. C. Greubel, Sep 25 2019

Maple

seq(coeff(series((1+t)*(1-t^9)/(1-11*t+65*t^9-55*t^10), t, n+1), t, n), n = 0..30); # G. C. Greubel, Sep 25 2019

Mathematica

CoefficientList[Series[(1+t)*(1-t^9)/(1-11*t+65*t^9-55*t^10), {t,0,30}], t] (* or *) coxG[{9, 55, -10}] (* The coxG program is at A169452 *) (* G. C. Greubel, Sep 25 2019 *)

PARI

my(t='t+O('t^30)); Vec((1+t)*(1-t^9)/(1-11*t+65*t^9-55*t^10)) \\ G. C. Greubel, Sep 25 2019

Sage

def A165266_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P((1+t)*(1-t^9)/(1-11*t+65*t^9-55*t^10)).list()
A165266_list(30) # G. C. Greubel, Sep 25 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)/(55*t^9 - 10*t^8 - 10*t^7 - 10*t^6 - 10*t^5 - 10*t^4 - 10*t^3 - 10*t^2 - 10*t + 1).