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A166442 Number of reduced words of length n in Coxeter group on 48 generators S_i with relations (S_i)^2 = (S_i S_j)^11 = I.

%I A166442 #27 Jul 27 2024 09:41:02
%S A166442 1,48,2256,106032,4983504,234224688,11008560336,517402335792,
%T A166442 24317909782224,1142941759764528,53718262708932816,
%U A166442 2524758347319841224,118663642324032484512,5577191189229524281440,262127985893787524168352
%N A166442 Number of reduced words of length n in Coxeter group on 48 generators S_i with relations (S_i)^2 = (S_i S_j)^11 = I.
%C A166442 The initial terms coincide with those of A170767, although the two sequences are eventually different.
%C A166442 Computed with MAGMA using commands similar to those used to compute A154638.
%H A166442 G. C. Greubel, <a href="/A166442/b166442.txt">Table of n, a(n) for n = 0..500</a>
%H A166442 <a href="/index/Rec#order_11">Index entries for linear recurrences with constant coefficients</a>, signature (46,46,46,46,46,46,46,46,46,46,-1081).
%F A166442 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)/(1081*t^11 - 46*t^10 - 46*t^9 - 46*t^8 - 46*t^7 - 46*t^6 - 46*t^5 - 46*t^4 - 46*t^3 - 46*t^2 - 46*t + 1).
%F A166442 From _G. C. Greubel_, Jul 26 2024: (Start)
%F A166442 a(n) = 46*Sum_{j=1..10} a(n-j) - 1081*a(n-11).
%F A166442 G.f.: (1+x)*(1-x^11)/(1 - 47*x + 1127*x^11 - 1081*x^12). (End)
%t A166442 With[{num=Total[2t^Range[10] ]+t^11+1,den=Total[-46 t^Range[10]]+ 1081t^11+ 1}, CoefficientList[Series[num/den,{t,0,30}],t]] (* _Harvey P. Dale_, Jul 21 2011 *)
%t A166442 CoefficientList[Series[(x^11 + 2 x^10 + 2 x^9 + 2 x^8 + 2 x^7 + 2 x^6 + 2 x^5 + 2 x^4 + 2 x^3 + 2 x^2 + 2 x + 1)/(1081 x^11 - 46 x^10 - 46 x^9 - 46 x^8 - 46 x^7 - 46 x^6 - 46 x^5 - 46 x^4 - 46 x^3 - 46 x^2 - 46 x + 1), {x, 0, 20}], x] (* _Vincenzo Librandi_, May 10 2015 *)
%t A166442 coxG[{11, 1081, -46, 30}] (* The coxG program is at A169452 *) (* _G. C. Greubel_, Jul 26 2024 *)
%o A166442 (Magma)
%o A166442 R<x>:=PowerSeriesRing(Integers(), 30);
%o A166442 f:= func< p,q,x | (1+x)*(1-x^11)/(1-(q+1)*x+(p+q)*x^11-p*x^12) >;
%o A166442 Coefficients(R!( f(1081,46,x) )); // _G. C. Greubel_, Jul 26 2024
%o A166442 (SageMath)
%o A166442 def f(p,q,x): return (1+x)*(1-x^11)/(1-(q+1)*x+(p+q)*x^11-p*x^12)
%o A166442 def A166442_list(prec):
%o A166442     P.<x> = PowerSeriesRing(ZZ, prec)
%o A166442     return P( f(1081,46,x) ).list()
%o A166442 A166442_list(30) # _G. C. Greubel_, Jul 26 2024
%Y A166442 Cf. A154638, A169452, A170767.
%K A166442 nonn
%O A166442 0,2
%A A166442 _John Cannon_ and _N. J. A. Sloane_, Dec 03 2009