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A166557 Number of reduced words of length n in Coxeter group on 12 generators S_i with relations (S_i)^2 = (S_i S_j)^12 = I.

%I A166557 #16 Mar 19 2025 07:22:46
%S A166557 1,12,132,1452,15972,175692,1932612,21258732,233846052,2572306572,
%T A166557 28295372292,311249095212,3423740047266,37661140519200,
%U A166557 414272545703280,4556998002648960,50126978028180240,551396758299441120,6065364341177895600,66719007751681327680
%N A166557 Number of reduced words of length n in Coxeter group on 12 generators S_i with relations (S_i)^2 = (S_i S_j)^12 = I.
%C A166557 The initial terms coincide with those of A003954, although the two sequences are eventually different.
%C A166557 Computed with MAGMA using commands similar to those used to compute A154638.
%H A166557 G. C. Greubel, <a href="/A166557/b166557.txt">Table of n, a(n) for n = 0..500</a>
%H A166557 <a href="/index/Rec#order_12">Index entries for linear recurrences with constant coefficients</a>, signature (10,10,10,10,10,10,10,10,10,10,10,-55).
%F A166557 G.f.: (t^12 + 2*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)/(55*t^12 - 10*t^11 - 10*t^10 - 10*t^9 - 10*t^8 - 10*t^7 - 10*t^6 - 10*t^5 - 10*t^4 - 10*t^3 - 10*t^2 - 10*t +1).
%F A166557 From _G. C. Greubel_, Dec 03 2024: (Start)
%F A166557 a(n) = 10*Sum_{j=1..11} a(n-j) - 55*a(n-12).
%F A166557 G.f.: (1+t)*(1 - t^12)/(1 - 11*t + 65*t^12 - 55*t^13). (End)
%t A166557 CoefficientList[Series[(1+t)*(1-t^12)/(1-11*t+65*t^12-55*t^13), {t,0,50}], t]
%t A166557 (* _G. C. Greubel_, May 17 2016; Dec 03 2024 *)
%t A166557 coxG[{12,55,-10}] (* The coxG program is at A169452 *) (* _G. C. Greubel_, Dec 03 2024 *)
%o A166557 (Magma)
%o A166557 R<x>:=PowerSeriesRing(Integers(), 40);
%o A166557 Coefficients(R!( (1+x)*(1-x^12)/(1-11*x+65*x^12-55*x^13) )); // _G. C. Greubel_, Dec 03 2024
%o A166557 (SageMath)
%o A166557 def A166557_list(prec):
%o A166557     P.<x> = PowerSeriesRing(ZZ, prec)
%o A166557     return P( (1+x)*(1-x^12)/(1-11*x+65*x^12-55*x^13) ).list()
%o A166557 A166557_list(40) # _G. C. Greubel_, Dec 03 2024
%Y A166557 Cf. A003954, A154638, A169452.
%K A166557 nonn
%O A166557 0,2
%A A166557 _John Cannon_ and _N. J. A. Sloane_, Dec 03 2009