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A166412 Number of reduced words of length n in Coxeter group on 18 generators S_i with relations (S_i)^2 = (S_i S_j)^11 = I.

%I A166412 #15 Jul 25 2024 02:35:16
%S A166412 1,18,306,5202,88434,1503378,25557426,434476242,7386096114,
%T A166412 125563633938,2134581776946,36287890207929,616894133532192,
%U A166412 10487200270003200,178282404589305312,3030800878005455808,51523614925876262304
%N A166412 Number of reduced words of length n in Coxeter group on 18 generators S_i with relations (S_i)^2 = (S_i S_j)^11 = I.
%C A166412 The initial terms coincide with those of A170737, although the two sequences are eventually different.
%C A166412 Computed with MAGMA using commands similar to those used to compute A154638.
%H A166412 G. C. Greubel, <a href="/A166412/b166412.txt">Table of n, a(n) for n = 0..500</a>
%H A166412 <a href="/index/Rec#order_11">Index entries for linear recurrences with constant coefficients</a>, signature (16,16,16,16,16,16,16,16,16,16,-136).
%F A166412 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)/(136*t^11 - 16*t^10 - 16*t^9 - 16*t^8 - 16*t^7 - 16*t^6 - 16*t^5 - 16*t^4 - 16*t^3 - 16*t^2 - 16*t + 1).
%F A166412 From _G. C. Greubel_, Jul 23 2024: (Start)
%F A166412 a(n) = 16*Sum_{j=1..10} a(n-j) - 136*a(n-11).
%F A166412 G.f.: (1+x)*(1 - x^11)/(1 - 17*x + 152*x^11 - 136*x^12). (End)
%t A166412 CoefficientList[Series[(1+t)*(1-t^11)/(1-17*t+152*t^11-136*t^12), {t, 0,50}], t] (* _G. C. Greubel_, May 12 2016; Jul 23 2024 *)
%t A166412 coxG[{11, 136, -16, 30}] (* The coxG program is at A169452 *)(* _G. C. Greubel_, Jul 23 2024 *)
%o A166412 (Magma)
%o A166412 R<x>:=PowerSeriesRing(Integers(), 30);
%o A166412 Coefficients(R!( (1+x)*(1-x^11)/(1-17*x+152*x^11-136*x^12) )); // _G. C. Greubel_, Jul 23 2024
%o A166412 (SageMath)
%o A166412 def A166412_list(prec):
%o A166412     P.<x> = PowerSeriesRing(ZZ, prec)
%o A166412     return P( (1+x)*(1-x^11)/(1-17*x+152*x^11-136*x^12) ).list()
%o A166412 A166412_list(30) # _G. C. Greubel_, Jul 23 2024
%Y A166412 Cf. A154638, A169452, A170737.
%K A166412 nonn
%O A166412 0,2
%A A166412 _John Cannon_ and _N. J. A. Sloane_, Dec 03 2009