A165512 Number of reduced words of length n in Coxeter group on 29 generators S_i with relations (S_i)^2 = (S_i S_j)^9 = I.

1, 29, 812, 22736, 636608, 17825024, 499100672, 13974818816, 391294926848, 10956257951338, 306775222626096, 8589706233212790, 240511774521056976, 6734329686340363296, 188561231210551675392, 5279714473700048997888

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..689
• Index entries for linear recurrences with constant coefficients, signature (27,27,27,27,27,27,27,27,-378).

Programs

GAP

a:=[29, 812, 22736, 636608, 17825024, 499100672, 13974818816, 391294926848, 10956257951338];; for n in [10..20] do a[n]:=27*Sum([1..8], j-> a[n-j]) -378*a[n-9]; od; Concatenation([1], a); # G. C. Greubel, Sep 16 2019

Magma

R:=PowerSeriesRing(Integers(), 20); Coefficients(R!( (1+t)*(1-t^9)/(1-28*t+405*t^9-378*t^10) )); // G. C. Greubel, Oct 21 2018

Maple

seq(coeff(series((1+t)*(1-t^9)/(1-28*t+405*t^9-378*t^10), t, n+1), t, n), n = 0..20); # G. C. Greubel, Sep 16 2019

Mathematica

CoefficientList[Series[(1+t)*(1-t^9)/(1-28*t+405*t^9-378*t^10), {t,0, 20}], t] (* G. C. Greubel, Oct 21 2018 *)
coxG[{9, 378, -27}] (* The coxG program is at A169452 *) (* G. C. Greubel, Sep 16 2019 *)

PARI

my(t='t+O('t^20)); Vec((1+t)*(1-t^9)/(1-28*t+405*t^9-378*t^10)) \\ G. C. Greubel, Oct 21 2018

Sage

def A165512_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P((1+t)*(1-t^9)/(1-28*t+405*t^9-378*t^10)).list()
A165512_list(20) # G. C. Greubel, Sep 16 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)/(378*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).