%I #16 Jul 25 2024 14:48:42

%S 1,27,702,18252,474552,12338352,320797152,8340725952,216858874752,

%T 5638330743552,146596599332352,3811511582640801,99099301148651700,

%U 2576581829864707275,66991127576476229100,1741769316988221795300

%N 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.

%C The initial terms coincide with those of A170746, although the two sequences are eventually different.

%C Computed with MAGMA using commands similar to those used to compute A154638.

%H G. C. Greubel, <a href="/A166421/b166421.txt">Table of n, a(n) for n = 0..500</a>

%H <a href="/index/Rec#order_11">Index entries for linear recurrences with constant coefficients</a>, signature (25,25,25,25,25,25,25,25,25,25,-325).

%F 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).

%F From _G. C. Greubel_, Jan 17 2023: (Start)

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

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

%t 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 *)

%t coxG[{11,325,-25}] (* The coxG program is at A169452 *) (* _Harvey P. Dale_, May 22 2021 *)

%o (Magma)

%o R<x>:=PowerSeriesRing(Integers(), 30);

%o Coefficients(R!( (1+x)*(1-x^11)/(1-26*x+350*x^11-325*x^12) )); // _G. C. Greubel_, Jul 25 2024

%o (SageMath)

%o def A166421_list(prec):

%o P.<x> = PowerSeriesRing(ZZ, prec)

%o return P( (1+x)*(1-x^11)/(1-26*x+350*x^11-325*x^12) ).list()

%o A166421_list(30) # _G. C. Greubel_, Jul 25 2024

%Y Cf. A154638, A169452, A170746.

%K nonn

%O 0,2

%A _John Cannon_ and _N. J. A. Sloane_, Dec 03 2009