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A166413 Number of reduced words of length n in Coxeter group on 19 generators S_i with relations (S_i)^2 = (S_i S_j)^11 = I.

%I A166413 #19 Jul 23 2024 20:29:57
%S A166413 1,19,342,6156,110808,1994544,35901792,646232256,11632180608,
%T A166413 209379250944,3768826516992,67838877305685,1221099791499252,
%U A166413 21979796246931303,395636332443769260,7121453983969951188
%N A166413 Number of reduced words of length n in Coxeter group on 19 generators S_i with relations (S_i)^2 = (S_i S_j)^11 = I.
%C A166413 The initial terms coincide with those of A170738, although the two sequences are eventually different.
%C A166413 Computed with MAGMA using commands similar to those used to compute A154638.
%H A166413 G. C. Greubel, <a href="/A166413/b166413.txt">Table of n, a(n) for n = 0..500</a>
%H A166413 <a href="/index/Rec#order_11">Index entries for linear recurrences with constant coefficients</a>, signature (17,17,17,17,17,17,17,17,17,17,-153).
%F A166413 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)/(153*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 A166413 From _G. C. Greubel_, Jul 23 2024: (Start)
%F A166413 a(n) = 17*Sum_{j=1..10} a(n-j) - 153*a(n-11).
%F A166413 G.f.: (1+x)*(1-x^11)/(1 - 18*x + 170*x^11 - 153*x^12). (End)
%t A166413 With[{p=153, q=17}, 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 12 2016; Jul 23 2024 *)
%t A166413 coxG[{11,153,-17}] (* The coxG program is at A169452 *) (* _Harvey P. Dale_, Nov 26 2022 *)
%o A166413 (Magma)
%o A166413 R<x>:=PowerSeriesRing(Integers(), 30);
%o A166413 Coefficients(R!( (1+x)*(1-x^11)/(1-18*x+170*x^11-153*x^12) )); // _G. C. Greubel_, Jul 23 2024
%o A166413 (SageMath)
%o A166413 def A166413_list(prec):
%o A166413     P.<x> = PowerSeriesRing(ZZ, prec)
%o A166413     return P( (1+x)*(1-x^11)/(1-18*x+170*x^11-153*x^12) ).list()
%o A166413 A166413_list(30) # _G. C. Greubel_, Jul 23 2024
%Y A166413 Cf. A154638, A169452, A170738.
%K A166413 nonn
%O A166413 0,2
%A A166413 _John Cannon_ and _N. J. A. Sloane_, Dec 03 2009