A166004 Number of reduced words of length n in Coxeter group on 29 generators S_i with relations (S_i)^2 = (S_i S_j)^10 = I.

1, 29, 812, 22736, 636608, 17825024, 499100672, 13974818816, 391294926848, 10956257951744, 306775222648426, 8589706234144560, 240511774555729782, 6734329687551532752, 188561231251193685024, 5279714475026444683776

Offset: 0

Author

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

Keywords

• nonn

Comments

The initial terms coincide with those of A170748, 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 (27,27,27,27,27,27,27,27,27,-378).

Programs

GAP

a:=[29, 812, 22736, 636608, 17825024, 499100672, 13974818816, 391294926848, 10956257951744, 306775222648426];; for n in [11..30] do a[n]:=27*Sum([1..9], j-> a[n-j]) - 378*a[n-10]; od; Concatenation([1], a); # G. C. Greubel, Oct 25 2019

Magma

R:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1+t)*(1-t^10)/(1-28*t+405*t^10-378*t^11) )); // G. C. Greubel, Oct 25 2019

Maple

seq(coeff(series((1+t)*(1-t^10)/(1-28*t+405*t^10-378*t^11), t, n+1), t, n), n = 0..30); # G. C. Greubel, Oct 25 2019

Mathematica

CoefficientList[Series[(1+t)*(1-t^10)/(1-28*t+405*t^10-378*t^11), {t, 0, 30}], t] (* G. C. Greubel, Apr 21 2016 *)
coxG[{10, 378, -27}] (* The coxG program is at A169452 *) (* G. C. Greubel, Oct 25 2019 *)

PARI

my(t='t+O('t^30)); Vec((1+t)*(1-t^10)/(1-28*t+405*t^10-378*t^11)) \\ G. C. Greubel, Oct 25 2019

Sage

def A166004_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P((1+t)*(1-t^10)/(1-28*t+405*t^10-378*t^11)).list()
A166004_list(30) # G. C. Greubel, Oct 25 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)/(378*t^10 - 27*t^9 - 27*t^8 - 27*t^7 - 27*t^6 - 27*t^5 - 27*t^4 - 27*t^3 - 27*t^2 - 27*t + 1).