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A166600 Number of reduced words of length n in Coxeter group on 19 generators S_i with relations (S_i)^2 = (S_i S_j)^12 = I.

%I A166600 #16 Dec 08 2024 03:59:46
%S A166600 1,19,342,6156,110808,1994544,35901792,646232256,11632180608,
%T A166600 209379250944,3768826516992,67838877305856,1221099791505237,
%U A166600 21979796247091188,395636332447586151,7121453984055556524
%N A166600 Number of reduced words of length n in Coxeter group on 19 generators S_i with relations (S_i)^2 = (S_i S_j)^12 = I.
%C A166600 The initial terms coincide with those of A170738, although the two sequences are eventually different.
%C A166600 Computed with MAGMA using commands similar to those used to compute A154638.
%H A166600 G. C. Greubel, <a href="/A166600/b166600.txt">Table of n, a(n) for n = 0..500</a>
%H A166600 <a href="/index/Rec#order_12">Index entries for linear recurrences with constant coefficients</a>, signature (17,17,17,17,17,17,17,17,17,17,17,-153).
%F A166600 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)/(153*t^12 - 17*t^11 - 17*t^10 - 17*t^9 -17*t^8 -17*t^7 - 17*t^6 - 17*t^5 - 17*t^4 - 17*t^3 - 17*t^2 -17*t + 1).
%F A166600 From _G. C. Greubel_, Dec 08 2024: (Start)
%F A166600 a(n) = 17*Sum_{j=1..11} a(n-j) - 153*a(n-12).
%F A166600 G.f.: (1+x)*(1-x^12)/(1 - 18*x + 170*x^12 - 153*x^13). (End)
%t A166600 CoefficientList[Series[(1+t)*(1-t^12)/(1-18*t+170*t^12-153*t^13), {t, 0, 50}], t] (* _G. C. Greubel_, May 18 2016; Dec 08 2024 *)
%t A166600 coxG[{12,153,-17}] (* The coxG program is at A169452 *) (* _Harvey P. Dale_, Oct 05 2016 *)
%o A166600 (Magma)
%o A166600 R<x>:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (1+x)*(1-x^12)/(1 - 18*x+170*x^12-153*x^13) )); // _G. C. Greubel_, Dec 08 2024
%o A166600 (SageMath)
%o A166600 def A166600_list(prec):
%o A166600     P.<x> = PowerSeriesRing(ZZ, prec)
%o A166600     return P( (1+x)*(1-x^12)/(1-18*x+170*x^12-153*x^13) ).list()
%o A166600 print(A166600_list(40)) # _G. C. Greubel_, Dec 08 2024
%Y A166600 Cf. A154638, A169452, A170738.
%K A166600 nonn
%O A166600 0,2
%A A166600 _John Cannon_ and _N. J. A. Sloane_, Dec 03 2009