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A166558 Number of reduced words of length n in Coxeter group on 13 generators S_i with relations (S_i)^2 = (S_i S_j)^12 = I.

%I A166558 #16 Dec 03 2024 03:23:20
%S A166558 1,13,156,1872,22464,269568,3234816,38817792,465813504,5589762048,
%T A166558 67077144576,804925734912,9659108818866,115909305825456,
%U A166558 1390911669894318,16690940038597968,200291280461569440,2403495365519559168,28841944386003420672,346103332629265575936
%N A166558 Number of reduced words of length n in Coxeter group on 13 generators S_i with relations (S_i)^2 = (S_i S_j)^12 = I.
%C A166558 The initial terms coincide with those of A170732, although the two sequences are eventually different.
%C A166558 Computed with MAGMA using commands similar to those used to compute A154638.
%H A166558 G. C. Greubel, <a href="/A166558/b166558.txt">Table of n, a(n) for n = 0..500</a>
%H A166558 <a href="/index/Rec#order_12">Index entries for linear recurrences with constant coefficients</a>, signature (11,11,11,11,11,11,11,11,11,11,11,-66).
%F A166558 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)/(66*t^12 - 11*t^11 - 11*t^10 - 11*t^9 - 11*t^8 - 11*t^7 - 11*t^6 - 11*t^5 - 11*t^4 - 11*t^3 - 11*t^2 - 11*t +1).
%F A166558 From _G. C. Greubel_, Dec 03 2024: (Start)
%F A166558 a(n) = 11*Sum_{j=1..11} a(n-j) - 66*a(n-12).
%F A166558 G.f.: (1+x)*(1-x^12)/(1 - 12*x + 77*x^12 - 66*x^13). (End)
%t A166558 CoefficientList[Series[(1+t)*(1-t^12)/(1-12*t+77*t^12-66*t^13), {t,0,50}], t] (* _G. C. Greubel_, May 17 2016; Dec 03 2024 *)
%t A166558 coxG[{12,66,-11}] (* The coxG program is at A169452 *) (* _G. C. Greubel_, Dec 03 2024 *)
%o A166558 (Magma)
%o A166558 R<x>:=PowerSeriesRing(Integers(), 40);
%o A166558 Coefficients(R!( (1+x)*(1-x^12)/(1-12*x+77*x^12-66*x^13) )); // _G. C. Greubel_, Dec 03 2024
%o A166558 (SageMath)
%o A166558 def A166558_list(prec):
%o A166558     P.<x> = PowerSeriesRing(ZZ, prec)
%o A166558     return P( (1+x)*(1-x^12)/(1-12*x+77*x^12-66*x^13) ).list()
%o A166558 A166558_list(40) # _G. C. Greubel_, Dec 03 2024
%Y A166558 Cf. A154638, A169452, A170732.
%K A166558 nonn
%O A166558 0,2
%A A166558 _John Cannon_ and _N. J. A. Sloane_, Dec 03 2009