login
Number of reduced words of length n in Coxeter group on 10 generators S_i with relations (S_i)^2 = (S_i S_j)^12 = I.
1

%I #14 Aug 24 2024 02:06:03

%S 1,10,90,810,7290,65610,590490,5314410,47829690,430467210,3874204890,

%T 34867844010,313810596045,2824295364000,25418658272400,

%U 228767924419200,2058911319481200,18530201872706400,166771816830738000

%N Number of reduced words of length n in Coxeter group on 10 generators S_i with relations (S_i)^2 = (S_i S_j)^12 = I.

%C The initial terms coincide with those of A003952, 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="/A166543/b166543.txt">Table of n, a(n) for n = 0..500</a>

%H <a href="/index/Rec#order_12">Index entries for linear recurrences with constant coefficients</a>, signature (8,8,8,8,8,8,8,8,8,8,8,-36).

%F G.f.: (t^12 + 2*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)/(36*t^12 - 8*t^11 - 8*t^10 - 8*t^9 - 8*t^8 - 8*t^7 - 8*t^6 - 8*t^5 - 8*t^4 - 8*t^3 - 8*t^2 - 8*t + 1).

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

%F a(n) = 8*Sum_{j=1..11} a(n-j) - 36*a(n-12).

%F G.f.: (1+x)*(1-x^12)/(1 - 9*x + 44*x^12 - 36*x^13). (End)

%t CoefficientList[Series[(1+t)*(1-t^12)/(1-9*t+44*t^12-36*t^13), {t, 0, 50}], t] (* _G. C. Greubel_, May 16 2016; Aug 23 2024 *)

%t coxG[{12,36,-8,30}] (* The coxG program is at A169452 *) (* _G. C. Greubel_, Aug 23 2024 *)

%o (Magma)

%o R<x>:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1+x)*(1-x^12)/(1-9*x+44*x^12-36*x^13) )); // _G. C. Greubel_, Aug 23 2024

%o (SageMath)

%o def A166543_list(prec):

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

%o return P( (1+x)*(1-x^12)/(1-9*x+44*x^12-36*x^13) ).list()

%o A166543_list(30) # _G. C. Greubel_, Aug 23 2024

%Y Cf. A003952, A154638, A169452.

%K nonn

%O 0,2

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