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A166431 Number of reduced words of length n in Coxeter group on 37 generators S_i with relations (S_i)^2 = (S_i S_j)^11 = I.

%I A166431 #13 Jul 25 2024 23:05:53
%S A166431 1,37,1332,47952,1726272,62145792,2237248512,80540946432,
%T A166431 2899474071552,104381066575872,3757718396731392,135277862282329446,
%U A166431 4870003042163836080,175320109517897236410,6311523942644269461840
%N A166431 Number of reduced words of length n in Coxeter group on 37 generators S_i with relations (S_i)^2 = (S_i S_j)^11 = I.
%C A166431 The initial terms coincide with those of A170756, although the two sequences are eventually different.
%C A166431 Computed with MAGMA using commands similar to those used to compute A154638.
%H A166431 G. C. Greubel, <a href="/A166431/b166431.txt">Table of n, a(n) for n = 0..500</a>
%H A166431 <a href="/index/Rec#order_11">Index entries for linear recurrences with constant coefficients</a>, signature (35,35,35,35,35,35,35,35,35,35,-630).
%F A166431 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)/(630*t^11 - 35*t^10 - 35*t^9 - 35*t^8 - 35*t^7 - 35*t^6 - 35*t^5 - 35*t^4 - 35*t^3 - 35*t^2 - 35*t + 1).
%F A166431 From _G. C. Greubel_, Jul 25 2024: (Start)
%F A166431 a(n) = 35*Sum_{j=1..10} a(n-j) - 630*a(n-11).
%F A166431 G.f.: (1+x)*(1-x^11)/(1 - 36*x + 665*x^11 - 630*x^12). (End)
%t A166431 With[{p=630, q=35}, 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 14 2016; Jul 25 2024 *)
%t A166431 coxG[{11, 630, -35. 30}] (* The coxG program is at A169452 *) (* _G. C. Greubel_, Jul 25 2024 *)
%o A166431 (Magma)
%o A166431 R<x>:=PowerSeriesRing(Integers(), 30);
%o A166431 Coefficients(R!( (1+x)*(1-x^11)/(1-36*x+665*x^11-630*x^12) )); // _G. C. Greubel_, Jul 25 2024
%o A166431 (SageMath)
%o A166431 def A166431_list(prec):
%o A166431     P.<x> = PowerSeriesRing(ZZ, prec)
%o A166431     return P( (1+x)*(1-x^11)/(1-36*x+665*x^11-630*x^12) ).list()
%o A166431 A166431_list(30) # _G. C. Greubel_, Jul 25 2024
%Y A166431 Cf. A154638, A169452, A170756.
%K A166431 nonn
%O A166431 0,2
%A A166431 _John Cannon_ and _N. J. A. Sloane_, Dec 03 2009