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A166585 Number of reduced words of length n in Coxeter group on 17 generators S_i with relations (S_i)^2 = (S_i S_j)^12 = I.

%I A166585 #13 Dec 04 2024 05:50:52
%S A166585 1,17,272,4352,69632,1114112,17825792,285212672,4563402752,
%T A166585 73014444032,1168231104512,18691697672192,299067162754936,
%U A166585 4785074604076800,76561193665194120,1224979098642551040,19599665578271938560
%N A166585 Number of reduced words of length n in Coxeter group on 17 generators S_i with relations (S_i)^2 = (S_i S_j)^12 = I.
%C A166585 The initial terms coincide with those of A170736, although the two sequences are eventually different.
%C A166585 Computed with MAGMA using commands similar to those used to compute A154638.
%H A166585 G. C. Greubel, <a href="/A166585/b166585.txt">Table of n, a(n) for n = 0..500</a>
%H A166585 <a href="/index/Rec#order_12">Index entries for linear recurrences with constant coefficients</a>, signature (15,15,15,15,15,15,15,15,15,15,15,-120).
%F A166585 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)/(120*t^12 - 15*t^11 - 15*t^10 - 15*t^9 -15*t^8 - 15*t^7 - 15*t^6 - 15*t^5 - 15*t^4 - 15*t^3 - 15*t^2 -15*t +1).
%F A166585 From _G. C. Greubel_, Dec 04 2024: (Start)
%F A166585 a(n) = 15*Sum_{j=1..11} a(n-j) - 120*a(n-12).
%F A166585 G.f.: (1+x)*(1-x^12)/(1 - 16*x + 135*x^12 - 120*x^13). (End)
%t A166585 CoefficientList[Series[(1+t)*(1-t^12)/(1-16*t+135*t^12-120*t^13), {t, 0, 50}], t] (* _G. C. Greubel_, May 17 2016; Dec 04 2024 *)
%t A166585 coxG[{16,1081,-46}] (* The coxG program is at A169452 *) (* _G. C. Greubel_, Dec 04 2024 *)
%o A166585 (Magma) R<x>:=PowerSeriesRing(Integers(), 40);
%o A166585 Coefficients(R!( (1+x)*(1-x^12)/(1-16*x+135*x^12-120*x^13) )); // _G. C. Greubel_, Dec 04 2024
%o A166585 (SageMath)
%o A166585 def A166585_list(prec):
%o A166585     P.<x> = PowerSeriesRing(ZZ, prec)
%o A166585     return P( (1+x)*(1-x^12)/(1-16*x+135*x^12-120*x^13) ).list()
%o A166585 A166585_list(40) # _G. C. Greubel_, Dec 04 2024
%Y A166585 Cf. A154638, A169452, A170736.
%K A166585 nonn
%O A166585 0,2
%A A166585 _John Cannon_ and _N. J. A. Sloane_, Dec 03 2009