A165881 Number of reduced words of length n in Coxeter group on 19 generators S_i with relations (S_i)^2 = (S_i S_j)^10 = I.

1, 19, 342, 6156, 110808, 1994544, 35901792, 646232256, 11632180608, 209379250944, 3768826516821, 67838877299700, 1221099791339367, 21979796243114412, 395636332358163924, 7121453982124831776

Offset: 0

Author

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

Keywords

• nonn

Comments

The initial terms coincide with those of A170738, 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 (17,17,17,17,17,17,17,17,17,-153).

Crossrefs

Cf. A154638, A170738.

Programs

GAP

a:=[19, 342, 6156, 110808, 1994544, 35901792, 646232256, 11632180608, 209379250944, 3768826516821];; for n in [11..20] do a[n]:=17*Sum([1..9], j-> a[n-j]) -153*a[n-10]; od; Concatenation([1], a); # G. C. Greubel, Sep 24 2019

Magma

R:=PowerSeriesRing(Integers(), 20); Coefficients(R!( (1+t)*(1-t^10)/(1-18*t+170*t^10-153*t^11) )); // G. C. Greubel, Sep 24 2019

Maple

seq(coeff(series((1+t)*(1-t^10)/(1-18*t+170*t^10-153*t^11), t, n+1), t, n), n = 0..20); # G. C. Greubel, Sep 24 2019

Mathematica

CoefficientList[Series[(1+t)*(1-t^10)/(1-18*t+170*t^10-153*t^11), {t, 0, 20}], t] (* G. C. Greubel, Apr 17 2016 *)
coxG[{10,153,-17}] (* The coxG program is at A169452 *) (* Harvey P. Dale, May 23 2017 *)

PARI

my(t='t+O('t^20)); Vec((1+t)*(1-t^10)/(1-18*t+170*t^10-153*t^11)) \\ G. C. Greubel, Sep 24 2019

Sage

def A163878_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P((1+t)*(1-t^10)/(1-18*t+170*t^10-153*t^11)).list()
A163878_list(20) # G. C. Greubel, Sep 24 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)/(153*t^10 - 17*t^9 - 17*t^8 - 17*t^7 - 17*t^6 - 17*t^5 - 17*t^4 - 17*t^3 - 17*t^2 - 17*t + 1).