cp's OEIS Frontend

A166584 Number of reduced words of length n in Coxeter group on 16 generators S_i with relations (S_i)^2 = (S_i S_j)^12 = I.

%I A166584 #13 Dec 04 2024 05:50:22
%S A166584 1,16,240,3600,54000,810000,12150000,182250000,2733750000,41006250000,
%T A166584 615093750000,9226406250000,138396093749880,2075941406246400,
%U A166584 31139121093669120,467086816404633600,7006302246063456000
%N A166584 Number of reduced words of length n in Coxeter group on 16 generators S_i with relations (S_i)^2 = (S_i S_j)^12 = I.
%C A166584 The initial terms coincide with those of A170735, although the two sequences are eventually different.
%C A166584 Computed with MAGMA using commands similar to those used to compute A154638.
%H A166584 G. C. Greubel, <a href="/A166584/b166584.txt">Table of n, a(n) for n = 0..500</a>
%H A166584 <a href="/index/Rec#order_12">Index entries for linear recurrences with constant coefficients</a>, signature (14,14,14,14,14,14,14,14,14,14,14,-105).
%F A166584 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)/(105*t^12 - 14*t^11 - 14*t^10 - 14*t^9 -14*t^8 - 14*t^7 - 14*t^6 - 14*t^5 - 14*t^4 - 14*t^3 - 14*t^2 -14*t +1).
%F A166584 From _G. C. Greubel_, Dec 04 2024: (Start)
%F A166584 a(n) = 14*Sum_{j=1..11} a(n-j) - 105*a(n-12).
%F A166584 G.f.: (1+x)*(1-x^12)/(1 - 15*x + 119*x^12 - 105*x^13). (End)
%t A166584 CoefficientList[Series[(1+t)*(1-t^12)/(1-15*t+119*t^12-105*t^13), {t,0,50}], t] (* _G. C. Greubel_, May 17 2016; Dec 04 2024 *)
%t A166584 coxG[{12,105,-14}] (* The coxG program is at A169452 *) (* _G. C. Greubel_, Dec 04 2024 *)
%o A166584 (Magma)
%o A166584 R<x>:=PowerSeriesRing(Integers(), 40);
%o A166584 Coefficients(R!( (1+x)*(1-x^12)/(1-15*x+119*x^12-105*x^13) )); // _G. C. Greubel_, Dec 04 2024
%o A166584 (SageMath)
%o A166584 def A166584_list(prec):
%o A166584     P.<x> = PowerSeriesRing(ZZ, prec)
%o A166584     return P( (1+x)*(1-x^12)/(1-15*x+119*x^12-105*x^13) ).list()
%o A166584 A166584_list(40) # _G. C. Greubel_, Dec 04 2024
%Y A166584 Cf. A154638, A169452, A170735.
%K A166584 nonn
%O A166584 0,2
%A A166584 _John Cannon_ and _N. J. A. Sloane_, Dec 03 2009