A166421 Number of reduced words of length n in Coxeter group on 27 generators S_i with relations (S_i)^2 = (S_i S_j)^11 = I.

1, 27, 702, 18252, 474552, 12338352, 320797152, 8340725952, 216858874752, 5638330743552, 146596599332352, 3811511582640801, 99099301148651700, 2576581829864707275, 66991127576476229100, 1741769316988221795300

Offset: 0

Author

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

Keywords

• nonn

Comments

The initial terms coincide with those of A170746, 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 (25,25,25,25,25,25,25,25,25,25,-325).

Crossrefs

Cf. A154638, A169452, A170746.

Programs

Magma

R:=PowerSeriesRing(Integers(), 30);
Coefficients(R!( (1+x)*(1-x^11)/(1-26*x+350*x^11-325*x^12) )); // G. C. Greubel, Jul 25 2024

Mathematica

With[{p=325, q=25}, CoefficientList[Series[(1+t)*(1-t^11)/(1-(q+1)*t + (p+q)*t^11-p*t^12), {t,0,40}], t]] (* G. C. Greubel, May 13 2016; Jul 25 2024 *)
coxG[{11,325,-25}] (* The coxG program is at A169452 *) (* Harvey P. Dale, May 22 2021 *)

SageMath

def A166421_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1+x)*(1-x^11)/(1-26*x+350*x^11-325*x^12) ).list()
A166421_list(30) # G. C. Greubel, Jul 25 2024

Formula

G.f.: (t^11 + 2*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)/(325*t^11 - 25*t^10 - 25*t^9 - 25*t^8 - 25*t^7 - 25*t^6 - 25*t^5 - 25*t^4 - 25*t^3 - 25*t^2 - 25*t + 1).

From G. C. Greubel, Jan 17 2023: (Start)

a(n) = 25*Sum_{j=1..10} a(n-j) - 325*a(n-11).

G.f.: (1+x)*(1-x^11)/(1 - 26*x + 350*x^11 - 325*x^12). (End)