A165873 Number of reduced words of length n in Coxeter group on 13 generators S_i with relations (S_i)^2 = (S_i S_j)^10 = I.

1, 13, 156, 1872, 22464, 269568, 3234816, 38817792, 465813504, 5589762048, 67077144498, 804925733040, 9659108785326, 115909305290064, 1390911661874592, 16690939923220992, 200291278847362560

Offset: 0

Author

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

Keywords

• nonn

Comments

The initial terms coincide with those of A170732, 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 (11,11,11,11,11,11,11,11,11,-66).

Programs

GAP

a:=[13, 156, 1872, 22464, 269568, 3234816, 38817792, 465813504, 5589762048, 67077144498];; for n in [7..30] do a[n]:=11*Sum([1..9], j-> a[n-j]) -66*a[n-10]; od; Concatenation([1], a); # G. C. Greubel, Sep 23 2019

Magma

R:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1+t)*(1-t^10)/(1-12*t+77*t^10-66*t^11) )); // G. C. Greubel, Sep 23 2019

Maple

seq(coeff(series((1+t)*(1-t^10)/(1-12*t+77*t^10-66*t^11), t, n+1), t, n), n = 0 .. 30); # G. C. Greubel, Sep 23 2019

Mathematica

coxG[{10,66,-11}] (* The coxG program is at A169452 *) (* Harvey P. Dale, Jun 12 2015 *)
CoefficientList[Series[(1+t)*(1-t^10)/(1-12*t+77*t^10-66*t^11), {t, 0, 30}], t] (* G. C. Greubel, Sep 23 2019 *)

PARI

my(t='t+O('t^30)); Vec((1+t)*(1-t^10)/(1-12*t+77*t^10-66*t^11)) \\ G. C. Greubel, Sep 23 2019

Sage

def A165873_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P((1+t)*(1-t^10)/(1-12*t+77*t^10-66*t^11)).list()
A165873_list(30) # 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)/(66*t^10 - 11*t^9 - 11*t^8 - 11*t^7 - 11*t^6 - 11*t^5 - 11*t^4 - 11*t^3 - 11*t^2 - 11*t + 1).