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A166435 Number of reduced words of length n in Coxeter group on 41 generators S_i with relations (S_i)^2 = (S_i S_j)^11 = I.

%I A166435 #14 Jul 26 2024 06:18:36
%S A166435 1,41,1640,65600,2624000,104960000,4198400000,167936000000,
%T A166435 6717440000000,268697600000000,10747904000000000,429916159999999180,
%U A166435 17196646399999934400,687865855999996064820,27514634239999790145600
%N A166435 Number of reduced words of length n in Coxeter group on 41 generators S_i with relations (S_i)^2 = (S_i S_j)^11 = I.
%C A166435 The initial terms coincide with those of A170760, although the two sequences are eventually different.
%C A166435 Computed with MAGMA using commands similar to those used to compute A154638.
%H A166435 G. C. Greubel, <a href="/A166435/b166435.txt">Table of n, a(n) for n = 0..500</a>
%H A166435 <a href="/index/Rec#order_11">Index entries for linear recurrences with constant coefficients</a>, signature (39,39,39,39,39,39,39,39,39,39,-780).
%F A166435 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)/(780*t^11 - 39*t^10 - 39*t^9 - 39*t^8 - 39*t^7 - 39*t^6 - 39*t^5 - 39*t^4 - 39*t^3 - 39*t^2 - 39*t + 1).
%F A166435 From _G. C. Greubel_, Jul 26 2024: (Start)
%F A166435 a(n) = 39*Sum_{j=1..10} a(n-j) - 780*a(n-11).
%F A166435 G.f.: (1+x)*(1-x^11)/(1 - 40*x + 819*x^11 - 780*x^12). (End)
%t A166435 With[{p=780, q=39}, 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 26 2024 *)
%t A166435 coxG[{11, 780, -39, 30}] (* The coxG program is at A169452 *) (* _G. C. Greubel_, Jul 26 2024 *)
%o A166435 (Magma)
%o A166435 R<x>:=PowerSeriesRing(Integers(), 30);
%o A166435 Coefficients(R!( (1+x)*(1-x^11)/(1-40*x+819*x^11-780*x^12) )); // _G. C. Greubel_, Jul 26 2024
%o A166435 (SageMath)
%o A166435 def A166435_list(prec):
%o A166435     P.<x> = PowerSeriesRing(ZZ, prec)
%o A166435     return P( (1+x)*(1-x^11)/(1-40*x+819*x^11-780*x^12) ).list()
%o A166435 A166435_list(30) # _G. C. Greubel_, Jul 26 2024
%Y A166435 Cf. A154638, A169452, A170760.
%K A166435 nonn
%O A166435 0,2
%A A166435 _John Cannon_ and _N. J. A. Sloane_, Dec 03 2009