A165874 Number of reduced words of length n in Coxeter group on 14 generators S_i with relations (S_i)^2 = (S_i S_j)^10 = I.

1, 14, 182, 2366, 30758, 399854, 5198102, 67575326, 878479238, 11420230094, 148462991131, 1930018883520, 25090245470472, 326173190917392, 4240251479342424, 55123269197863776, 716602499135588520

Offset: 0

Author

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

Keywords

• nonn

Comments

The initial terms coincide with those of A170733, 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 (12,12,12,12,12,12,12,12,12,-78).

Programs

GAP

a:=[14, 182, 2366, 30758, 399854, 5198102, 67575326, 878479238, 11420230094, 148462991131];; for n in [11..20] do a[n]:=12*Sum([1..9], j-> a[n-j]) -78*a[n-10]; od; Concatenation([1], a); # G. C. Greubel, Sep 23 2019

Magma

R:=PowerSeriesRing(Integers(), 20); Coefficients(R!( (1+t)*(1-t^10)/(1-13*t+90*t^10-78*t^11) )); // G. C. Greubel, Sep 23 2019

Maple

seq(coeff(series((1+t)*(1-t^10)/(1-13*t+90*t^10-78*t^11), t, n+1), t, n), n = 0..20); # G. C. Greubel, Sep 23 2019

Mathematica

CoefficientList[Series[(1+t)*(1-t^10)/(1-13*t+90*t^10-78*t^11), {t, 0, 25}], t] (* G. C. Greubel, Apr 17 2016 *)
coxG[{10, 78, -12}] (* The coxG program is at A169452 *) (* G. C. Greubel, Sep 23 2019 *)

PARI

my(t='t+O('t^20)); Vec((1+t)*(1-t^10)/(1-13*t+90*t^10-78*t^11)) \\ G. C. Greubel, Sep 23 2019

Sage

def A165874_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P((1+t)*(1-t^10)/(1-13*t+90*t^10-78*t^11)).list()
A165874_list(20) # G. C. Greubel, Sep 23 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)/(78*t^10 - 12*t^9 - 12*t^8 - 12*t^7 - 12*t^6 - 12*t^5 - 12*t^4 - 12*t^3 - 12*t^2 - 12*t + 1).