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A166369 Number of reduced words of length n in Coxeter group on 11 generators S_i with relations (S_i)^2 = (S_i S_j)^11 = I.

%I A166369 #22 Jul 23 2024 15:11:35
%S A166369 1,11,110,1100,11000,110000,1100000,11000000,110000000,1100000000,
%T A166369 11000000000,109999999945,1099999998900,10999999983555,
%U A166369 109999999781100,1099999997266500,10999999967220000,109999999617750000
%N A166369 Number of reduced words of length n in Coxeter group on 11 generators S_i with relations (S_i)^2 = (S_i S_j)^11 = I.
%C A166369 The initial terms coincide with those of A003953, although the two sequences are eventually different.
%C A166369 Computed with MAGMA using commands similar to those used to compute A154638.
%H A166369 G. C. Greubel, <a href="/A166369/b166369.txt">Table of n, a(n) for n = 0..500</a>
%H A166369 <a href="/index/Rec#order_11">Index entries for linear recurrences with constant coefficients</a>, signature (9,9,9,9,9,9,9,9,9,9,-45).
%F A166369 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)/(45*t^11 - 9*t^10 - 9*t^9 - 9*t^8 - 9*t^7 - 9*t^6 - 9*t^5 - 9*t^4 - 9*t^3 - 9*t^2 - 9*t + 1).
%F A166369 From _G. C. Greubel_, Jul 23 2024: (Start)
%F A166369 a(n) = 9*Sum_{j=1..10} a(n-j) - 45*a(n-11).
%F A166369 G.f.: (1+t)*(1-t^11)/(1 - 10*t + 54*t^11 - 45*t^12). (End)
%t A166369 With[{p=45, q=9}, 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 10 2016; Jul 23 2024 *)
%t A166369 coxG[{11,45,-9}] (* The coxG program is at A169452 *) (* _Harvey P. Dale_, Jun 12 2016 *)
%o A166369 (Magma)
%o A166369 R<x>:=PowerSeriesRing(Integers(), 30);
%o A166369 Coefficients(R!( (1+x)*(1-x^11)/(1-10*x+54*x^11-45*x^12) )); // _G. C. Greubel_, Jul 23 2024
%o A166369 (SageMath)
%o A166369 def A166369_list(prec):
%o A166369     P.<x> = PowerSeriesRing(ZZ, prec)
%o A166369     return P( (1+x)*(1-x^11)/(1-10*x+54*x^11-45*x^12) ).list()
%o A166369 A166369_list(30) # _G. C. Greubel_, Jul 23 2024
%Y A166369 Cf. A003953, A154638, A169452.
%K A166369 nonn
%O A166369 0,2
%A A166369 _John Cannon_ and _N. J. A. Sloane_, Dec 03 2009