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A166410 Number of reduced words of length n in Coxeter group on 16 generators S_i with relations (S_i)^2 = (S_i S_j)^11 = I.

%I A166410 #19 Jul 23 2024 18:54:43
%S A166410 1,16,240,3600,54000,810000,12150000,182250000,2733750000,41006250000,
%T A166410 615093750000,9226406249880,138396093746400,2075941406169120,
%U A166410 31139121092133600,467086816375956000,7006302245548620000
%N A166410 Number of reduced words of length n in Coxeter group on 16 generators S_i with relations (S_i)^2 = (S_i S_j)^11 = I.
%C A166410 The initial terms coincide with those of A170735, although the two sequences are eventually different.
%C A166410 Computed with MAGMA using commands similar to those used to compute A154638.
%H A166410 G. C. Greubel, <a href="/A166410/b166410.txt">Table of n, a(n) for n = 0..500</a>
%H A166410 <a href="/index/Rec#order_11">Index entries for linear recurrences with constant coefficients</a>, signature (14,14,14,14,14,14,14,14,14,14,-105).
%F A166410 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)/(105*t^11 - 14*t^10 - 14*t^9 - 14*t^8 - 14*t^7 - 14*t^6 - 14*t^5 - 14*t^4 - 14*t^3 - 14*t^2 - 14*t + 1).
%F A166410 From _G. C. Greubel_, Jul 23 2024: (Start)
%F A166410 a(n) = 14*Sum_{j=1..10} a(n-j) - 105*a(n-11).
%F A166410 G.f.: (1+x)*(1-x^11)/(1 - 15*x + 119*x^11 - 105*x^12). (End)
%t A166410 With[{p=105, q=14}, 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 A166410 coxG[{11,105,-14}] (* The coxG program is at A169452 *) (* _Harvey P. Dale_, May 24 2021 *)
%o A166410 (Magma)
%o A166410 R<x>:=PowerSeriesRing(Integers(), 30);
%o A166410 Coefficients(R!( (1+x)*(1-x^11)/(1-15*x+119*x^11-105*x^12) )); // _G. C. Greubel_, Jul 23 2024
%o A166410 (SageMath)
%o A166410 def A166410_list(prec):
%o A166410     P.<x> = PowerSeriesRing(ZZ, prec)
%o A166410     return P( (1+x)*(1-x^11)/(1-15*x+119*x^11-105*x^12) ).list()
%o A166410 A166410_list(30) # _G. C. Greubel_, Jul 23 2024
%Y A166410 Cf. A154638, A169452, A170735.
%K A166410 nonn
%O A166410 0,2
%A A166410 _John Cannon_ and _N. J. A. Sloane_, Dec 03 2009