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A166583 Number of reduced words of length n in Coxeter group on 15 generators S_i with relations (S_i)^2 = (S_i S_j)^12 = I.

%I A166583 #16 Dec 03 2024 08:20:38
%S A166583 1,15,210,2940,41160,576240,8067360,112943040,1581202560,22136835840,
%T A166583 309915701760,4338819824640,60743477544855,850408685626500,
%U A166583 11905721598750525,166680102382220700,2333521433347076700
%N A166583 Number of reduced words of length n in Coxeter group on 15 generators S_i with relations (S_i)^2 = (S_i S_j)^12 = I.
%C A166583 The initial terms coincide with those of A170734, although the two sequences are eventually different.
%C A166583 Computed with MAGMA using commands similar to those used to compute A154638.
%H A166583 G. C. Greubel, <a href="/A166583/b166583.txt">Table of n, a(n) for n = 0..500</a>
%H A166583 <a href="/index/Rec#order_12">Index entries for linear recurrences with constant coefficients</a>, signature (13,13,13,13,13,13,13,13,13,13,13,-91).
%F A166583 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)/(91*t^12 - 13*t^11 - 13*t^10 - 13*t^9 - 13*t^8 - 13*t^7 - 13*t^6 - 13*t^5 - 13*t^4 - 13*t^3 - 13*t^2 - 13*t +1).
%F A166583 From _G. C. Greubel_, Dec 03 2024: (Start)
%F A166583 a(n) = 13*Sum_{j=1..11} a(n-j) - 91*a(n-12).
%F A166583 G.f.: (1+x)*(1-x^12)/(1 - 14*x + 104*x^12 - 91*x^13). (End)
%t A166583 CoefficientList[Series[(1+x)*(1-x^12)/(1 - 14*x + 104*x^12 - 91*x^13), {t, 0, 50}], t] (* _G. C. Greubel_, May 17 2016; Dec 03 2024 *)
%t A166583 coxG[{12,91,-13}] (* The coxG program is at A169452 *) (* _G. C. Greubel_, Dec 03 2024 *)
%o A166583 (Magma)
%o A166583 R<x>:=PowerSeriesRing(Integers(), 40);
%o A166583 Coefficients(R!( (1+x)*(1-x^12)/(1-14*x+104*x^12-91*x^13) )); // _G. C. Greubel_, Dec 03 2024
%o A166583 (SageMath)
%o A166583 def A166583_list(prec):
%o A166583     P.<x> = PowerSeriesRing(ZZ, prec)
%o A166583     return P( (1+x)*(1-x^12)/(1-14*x+104*x^12-91*x^13) ).list()
%o A166583 A166583_list(40) # _G. C. Greubel_, Dec 03 2024
%Y A166583 Cf. A154638, A169452, A170734.
%K A166583 nonn
%O A166583 0,2
%A A166583 _John Cannon_ and _N. J. A. Sloane_, Dec 03 2009