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A166463 Number of reduced words of length n in Coxeter group on 50 generators S_i with relations (S_i)^2 = (S_i S_j)^11 = I.

%I A166463 #15 Jul 27 2024 03:18:38
%S A166463 1,50,2450,120050,5882450,288240050,14123762450,692064360050,
%T A166463 33911153642450,1661646528480050,81420679895522450,
%U A166463 3989613314880598825,195491052429149282400,9579061569028311897600,469374016882387138922400
%N A166463 Number of reduced words of length n in Coxeter group on 50 generators S_i with relations (S_i)^2 = (S_i S_j)^11 = I.
%C A166463 The initial terms coincide with those of A170769, although the two sequences are eventually different.
%C A166463 Computed with MAGMA using commands similar to those used to compute A154638.
%H A166463 G. C. Greubel, <a href="/A166463/b166463.txt">Table of n, a(n) for n = 0..500</a>
%H A166463 <a href="/index/Rec#order_11">Index entries for linear recurrences with constant coefficients</a>, signature (48,48,48,48,48,48,48,48,48,48,-1176).
%F A166463 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)/(1176*t^11 - 48*t^10 - 48*t^9 - 48*t^8 - 48*t^7 - 48*t^6 - 48*t^5 - 48*t^4 - 48*t^3 - 48*t^2 - 48*t + 1).
%F A166463 From _G. C. Greubel_, Jul 27 2024: (Start)
%F A166463 a(n) = 48*Sum_{j=1..10} a(n-j) - 1176*a(n-11).
%F A166463 G.f.: (1+x)*(1-x^11)/(1 - 49*x + 1224*x^11 - 1176*x^12). (End)
%t A166463 With[{p=1176, q=48}, 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 15 2016; Jul 27 2024 *)
%t A166463 coxG[{11,1176,-48}] (* The coxG program is at A169452 *) (* _Harvey P. Dale_, Apr 29 2018 *)
%o A166463 (Magma)
%o A166463 R<x>:=PowerSeriesRing(Integers(), 30);
%o A166463 f:= func< p,q,x | (1+x)*(1-x^11)/(1-(q+1)*x+(p+q)*x^11-p*x^12) >;
%o A166463 Coefficients(R!( f(1176,48,x) )); // _G. C. Greubel_, Jul 27 2024
%o A166463 (SageMath)
%o A166463 def f(p,q,x): return (1+x)*(1-x^11)/(1-(q+1)*x+(p+q)*x^11-p*x^12)
%o A166463 def A166463_list(prec):
%o A166463     P.<x> = PowerSeriesRing(ZZ, prec)
%o A166463     return P( f(1176,48,x) ).list()
%o A166463 A166463_list(30) # _G. C. Greubel_, Jul 27 2024
%Y A166463 Cf. A154638, A169452, A170769.
%K A166463 nonn
%O A166463 0,2
%A A166463 _John Cannon_ and _N. J. A. Sloane_, Dec 03 2009