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A166601 Number of reduced words of length n in Coxeter group on 20 generators S_i with relations (S_i)^2 = (S_i S_j)^12 = I.

%I A166601 #16 Dec 30 2024 03:31:56
%S A166601 1,20,380,7220,137180,2606420,49521980,940917620,17877434780,
%T A166601 339671260820,6453753955580,122621325156020,2329805177964190,
%U A166601 44266298381316000,841059669244935600,15980133715652476800
%N A166601 Number of reduced words of length n in Coxeter group on 20 generators S_i with relations (S_i)^2 = (S_i S_j)^12 = I.
%C A166601 The initial terms coincide with those of A170739, although the two sequences are eventually different.
%C A166601 Computed with MAGMA using commands similar to those used to compute A154638.
%H A166601 G. C. Greubel, <a href="/A166601/b166601.txt">Table of n, a(n) for n = 0..500</a>
%H A166601 <a href="/index/Rec#order_12">Index entries for linear recurrences with constant coefficients</a>, signature (18,18,18,18,18,18,18,18,18,18,18,-171).
%F A166601 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)/(171*t^12 - 18*t^11 - 18*t^10 - 18*t^9 -18*t^8 -18*t^7 - 18*t^6 - 18*t^5 - 18*t^4 - 18*t^3 - 18*t^2 -18*t + 1).
%F A166601 From _G. C. Greubel_, Dec 30 2024: (Start)
%F A166601 a(n) = 18*Sum_{j=1..11} a(n-j) - 171*a(n-12).
%F A166601 G.f.: (1+x)*(1-x^12)/(1 - 19*x + 189*x^12 - 171*x^13). (End)
%t A166601 CoefficientList[Series[(1+t)*(1-t^12)/(1-19*t+189*t^12-171*t^13), {t,0,50}], t] (* _G. C. Greubel_, May 18 2016; Dec 30 2024 *)
%t A166601 coxG[{12,171,-18}] (* The coxG program is at A169452 *) (* _G. C. Greubel_, Dec 30 2024 *)
%o A166601 (Magma) R<x>:=PowerSeriesRing(Integers(), 50); Coefficients(R!( (1+x)*(1-x^12)/(1-19*x+189*x^12-171*x^13) )); // _G. C. Greubel_, Dec 30 2024
%o A166601 (SageMath)
%o A166601 def A166601_list(prec):
%o A166601     P.<x> = PowerSeriesRing(ZZ, prec)
%o A166601     return P( (1+x)*(1-x^12)/(1-19*x+189*x^12-171*x^13) ).list()
%o A166601 A166601_list(50) # _G. C. Greubel_, Dec 30 2024
%Y A166601 Cf. A154638, A169452, A170739.
%K A166601 nonn
%O A166601 0,2
%A A166601 _John Cannon_ and _N. J. A. Sloane_, Dec 03 2009