%I #14 Jul 25 2024 02:27:21

%S 1,21,420,8400,168000,3360000,67200000,1344000000,26880000000,

%T 537600000000,10752000000000,215039999999790,4300799999991600,

%U 86015999999748210,1720319999993288400,34406399999832252000

%N Number of reduced words of length n in Coxeter group on 21 generators S_i with relations (S_i)^2 = (S_i S_j)^11 = I.

%C The initial terms coincide with those of A170740, 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="/A166415/b166415.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 (19,19,19,19,19,19,19,19,19,19,-190).

%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)/(190*t^11 - 19*t^10 - 19*t^9 - 19*t^8 - 19*t^7 - 19*t^6 - 19*t^5 - 19*t^4 - 19*t^3 - 19*t^2 - 19*t + 1).

%F From _G. C. Greubel_, Jul 23 2024: (Start)

%F a(n) = 19*Sum_{j=1..10} a(n-j) - 190*a(n-11).

%F G.f.: (1+x)*(1-x^11)/(1 - 20*x + 209*x^11 - 190*x^12). (End)

%t CoefficientList[Series[(1+t)*(1-t^11)/(1-20*t+209*t^11-190*t^12), {t, 0, 50}], t] (* _G. C. Greubel_, May 13 2016; Jul 23 2024 *)

%t coxG[{11, 190, -19, 30}] (* The coxG program is at A169452 *) (* _G. C. Greubel_, Jul 23 2024 *)

%o (Magma)

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

%o Coefficients(R!( (1+x)*(1-x^11)/(1-20*x+209*x^11-190*x^12) )); // _G. C. Greubel_, Jul 23 2024

%o (SageMath)

%o def A166415_list(prec):

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

%o return P( (1+x)*(1-x^11)/(1-20*x+209*x^11-190*x^12) ).list()

%o A166415_list(30) # _G. C. Greubel_, Jul 23 2024

%Y Cf. A154638, A169452, A170740.

%K nonn

%O 0,2

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