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A166603 Number of reduced words of length n in Coxeter group on 21 generators S_i with relations (S_i)^2 = (S_i S_j)^12 = I.

%I A166603 #14 Jan 21 2025 23:59:30
%S A166603 1,21,420,8400,168000,3360000,67200000,1344000000,26880000000,
%T A166603 537600000000,10752000000000,215040000000000,4300799999999790,
%U A166603 86015999999991600,1720319999999748210,34406399999993288400
%N A166603 Number of reduced words of length n in Coxeter group on 21 generators S_i with relations (S_i)^2 = (S_i S_j)^12 = I.
%C A166603 The initial terms coincide with those of A170740, although the two sequences are eventually different.
%C A166603 Computed with MAGMA using commands similar to those used to compute A154638.
%H A166603 G. C. Greubel, <a href="/A166603/b166603.txt">Table of n, a(n) for n = 0..500</a>
%H A166603 <a href="/index/Rec#order_12">Index entries for linear recurrences with constant coefficients</a>, signature (19,19,19,19,19,19,19,19,19,19,19,-190).
%F A166603 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)/(190*t^12 - 19*t^11 - 19*t^10 - 19*t^9 -19*t^8 -19*t^7 - 19*t^6 - 19*t^5 - 19*t^4 - 19*t^3 - 19*t^2 - 19*t +1).
%F A166603 From _G. C. Greubel_, Jan 21 2025: (Start)
%F A166603 a(n) = 19*Sum_{j=1..11} a(n-j) - 190*a(n-12).
%F A166603 G.f.: (1+x)*(1-x^12)/(1 - 20*x + 209*x^12 - 190*x^13). (End)
%t A166603 CoefficientList[Series[(1+t)*(1-t^12)/(1-20*t+209*t^12-190*t^13), {t,0,50}], t] (* _G. C. Greubel_, May 18 2016; Jan 21 2025 *)
%t A166603 coxG[{12,190,-19}] (* The coxG program is at A169452 *) (* _G. C. Greubel_, Jan 21 2025 *)
%o A166603 (Magma) R<x>:=PowerSeriesRing(Integers(), 50); Coefficients(R!( (1+x)*(1-x^12)/(1-20*x+209*x^12-190*x^13) )); // _G. C. Greubel_, Jan 21 2025
%o A166603 (SageMath)
%o A166603 def A166603_list(prec):
%o A166603     P.<x> = PowerSeriesRing(ZZ, prec)
%o A166603     return P( (1+x)*(1-x^12)/(1-20*x+209*x^12-190*x^13) ).list()
%o A166603 A166603_list(50) # _G. C. Greubel_, Jan 21 2025
%Y A166603 Cf. A154638, A169452, A170740.
%K A166603 nonn
%O A166603 0,2
%A A166603 _John Cannon_ and _N. J. A. Sloane_, Dec 03 2009