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A166416 Number of reduced words of length n in Coxeter group on 22 generators S_i with relations (S_i)^2 = (S_i S_j)^11 = I.

%I A166416 #14 Jul 25 2024 02:31:02
%S A166416 1,22,462,9702,203742,4278582,89850222,1886854662,39623947902,
%T A166416 832102905942,17474161024782,366957381520191,7706105011919160,
%U A166416 161828205250200720,3398392310252080680,71366238515248871040
%N A166416 Number of reduced words of length n in Coxeter group on 22 generators S_i with relations (S_i)^2 = (S_i S_j)^11 = I.
%C A166416 The initial terms coincide with those of A170741, although the two sequences are eventually different.
%C A166416 Computed with MAGMA using commands similar to those used to compute A154638.
%H A166416 G. C. Greubel, <a href="/A166416/b166416.txt">Table of n, a(n) for n = 0..500</a>
%H A166416 <a href="/index/Rec#order_11">Index entries for linear recurrences with constant coefficients</a>, signature (20,20,20,20,20,20,20,20,20,20,-210).
%F A166416 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)/(210*t^11 - 20*t^10 - 20*t^9 - 20*t^8 - 20*t^7 - 20*t^6 - 20*t^5 - 20*t^4 - 20*t^3 - 20*t^2 - 20*t + 1).
%F A166416 From _G. C. Greubel_, Jul 23 2024: (Start)
%F A166416 a(n) = 20*Sum_{j=1..10} a(n-j) - 210*a(n-11).
%F A166416 G.f.: (1+x)*(1 - x^11)/(1 - 21*x + 230*x^11 - 210*x^12). (End)
%t A166416 With[{p=210, q=20}, 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 13 2016; Jul 23 2024 *)
%t A166416 coxG[{11, 210, -20, 30}] (* The coxG program is at A169452 *) (* _G. C. Greubel_, Jul 23 2024 *)
%o A166416 (Magma)
%o A166416 R<x>:=PowerSeriesRing(Integers(), 30);
%o A166416 Coefficients(R!( (1+x)*(1-x^11)/(1-21*x+230*x^11-210*x^12) )); // _G. C. Greubel_, Jul 23 2024
%o A166416 (SageMath)
%o A166416 def A166416_list(prec):
%o A166416     P.<x> = PowerSeriesRing(ZZ, prec)
%o A166416     return P( (1+x)*(1-x^11)/(1-21*x+230*x^11-210*x^12) ).list()
%o A166416 A166416_list(30) # _G. C. Greubel_, Jul 23 2024
%Y A166416 Cf. A154638, A169452, A170741.
%K A166416 nonn
%O A166416 0,2
%A A166416 _John Cannon_ and _N. J. A. Sloane_, Dec 03 2009