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A166414 Number of reduced words of length n in Coxeter group on 20 generators S_i with relations (S_i)^2 = (S_i S_j)^11 = I.

%I A166414 #20 Jul 25 2024 13:58:38
%S A166414 1,20,380,7220,137180,2606420,49521980,940917620,17877434780,
%T A166414 339671260820,6453753955580,122621325155830,2329805177957160,
%U A166414 44266298381117640,841059669239935560,15980133715534083240
%N A166414 Number of reduced words of length n in Coxeter group on 20 generators S_i with relations (S_i)^2 = (S_i S_j)^11 = I.
%C A166414 The initial terms coincide with those of A170739, although the two sequences are eventually different.
%C A166414 Computed with MAGMA using commands similar to those used to compute A154638.
%H A166414 G. C. Greubel, <a href="/A166414/b166414.txt">Table of n, a(n) for n = 0..500</a>
%H A166414 <a href="/index/Rec#order_11">Index entries for linear recurrences with constant coefficients</a>, signature (18,18,18,18,18,18,18,18,18,18,-171).
%F A166414 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)/(171*t^11 - 18*t^10 - 18*t^9 - 18*t^8 - 18*t^7 - 18*t^6 - 18*t^5 - 18*t^4 - 18*t^3 - 18*t^2 - 18*t + 1).
%F A166414 From _G. C. Greubel_, Jul 23 2024: (Start)
%F A166414 a(n) = 18*Sum_{j=1..10} a(n-j) - 171*a(n-11).
%F A166414 G.f.: (1+x)*(1 - x^11)/(1 - 19*x + 189*x^11 - 171*x^12). (End)
%t A166414 coxG[{11,171,-18}] (* The coxG program is at A169452 *) (* _Harvey P. Dale_, Sep 07 2015 *)
%t A166414 CoefficientList[Series[(1+t)*(1-t^11)/(1-19*t+189*t^11-171*t^12), {t, 0,50}], t] (* _G. C. Greubel_, May 12 2016; Jul 23 2024 *)
%o A166414 (Magma)
%o A166414 R<x>:=PowerSeriesRing(Integers(), 30);
%o A166414 Coefficients(R!( (1+x)*(1-x^11)/(1-19*x+189*x^11-171*x^12) )); // _G. C. Greubel_, Jul 23 2024
%o A166414 (SageMath)
%o A166414 def A166414_list(prec):
%o A166414     P.<x> = PowerSeriesRing(ZZ, prec)
%o A166414     return P( (1+x)*(1-x^11)/(1-19*x+189*x^11-171*x^12) ).list()
%o A166414 A166414_list(30) # _G. C. Greubel_, Jul 23 2024
%Y A166414 Cf. A154638, A169452, A170739.
%K A166414 nonn
%O A166414 0,2
%A A166414 _John Cannon_ and _N. J. A. Sloane_, Dec 03 2009