A166423 - OEIS

%I #14 Jul 25 2024 14:48:06

%S 1,29,812,22736,636608,17825024,499100672,13974818816,391294926848,

%T 10956257951744,306775222648832,8589706234166890,240511774556661552,

%U 6734329687586205558,188561231252404854480,5279714475067086693408

%N Number of reduced words of length n in Coxeter group on 29 generators S_i with relations (S_i)^2 = (S_i S_j)^11 = I.

%C The initial terms coincide with those of A170748, 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="/A166423/b166423.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 (27,27,27,27,27,27,27,27,27,27,-378).

%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)/(378*t^11 - 27*t^10 - 27*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).

%F From _G. C. Greubel_, Jul 25 2024: (Start)

%F a(n) = 27*Sum_{j=1..10} a(n-j) - 378*a(n-11).

%F G.f.: (1+x)*(1-x^11)/(1 - 28*x + 405*x^11 - 378*x^12). (End)

%t With[{p=378, q=27}, 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, 378, -27, 30}] (* The coxG program is at A169452 *) (* _G. C. Greubel_, Jul 25 2024 *)

%o (Magma)

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

%o Coefficients(R!( (1+x)*(1-x^11)/(1-28*x+405*x^11-378*x^12) )); // _G. C. Greubel_, Jul 25 2024

%o (SageMath)

%o def A166423_list(prec):

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

%o     return P( (1+x)*(1-x^11)/(1-28*x+405*x^11-378*x^12) ).list()

%o A166423_list(30) # _G. C. Greubel_, Jul 25 2024

%Y Cf. A154638, A169452, A170748.

%K nonn

%O 0,2

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