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A163220 Number of reduced words of length n in Coxeter group on 37 generators S_i with relations (S_i)^2 = (S_i S_j)^4 = I.

%I A163220 #14 Sep 08 2022 08:45:46
%S A163220 1,37,1332,47952,1725606,62097840,2234659770,80416702800,
%T A163220 2893883982570,104139615440700,3747579228757350,134860782963557700,
%U A163220 4853114416362432150,174644689291688511000,6284782282271390399250
%N A163220 Number of reduced words of length n in Coxeter group on 37 generators S_i with relations (S_i)^2 = (S_i S_j)^4 = I.
%C A163220 The initial terms coincide with those of A170756, although the two sequences are eventually different.
%C A163220 Computed with MAGMA using commands similar to those used to compute A154638.
%H A163220 G. C. Greubel, <a href="/A163220/b163220.txt">Table of n, a(n) for n = 0..640</a>
%H A163220 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (35, 35, 35, -630).
%F A163220 G.f.: (t^4 + 2*t^3 + 2*t^2 + 2*t + 1)/(630*t^4 - 35*t^3 - 35*t^2 - 35*t + 1).
%F A163220 a(n) = -630*a(n-4) + 35*Sum_{k=1..3} a(n-k). - _Wesley Ivan Hurt_, May 05 2021
%t A163220 CoefficientList[Series[(t^4+2*t^3+2*t^2+2*t+1)/(630*t^4-35*t^3-35*t^2 - 35*t+1), {t,0,20}], t] (* or *) LinearRecurrence[{35, 35, 35, -630}, {1, 37, 1332, 47952}, 20] (* _G. C. Greubel_, Dec 11 2016 *)
%t A163220 coxG[{4, 630, -35}] (* The coxG program is at A169452 *) (* _G. C. Greubel_, Apr 30 2019 *)
%o A163220 (PARI) my(t='t+O('t^20)); Vec((t^4+2*t^3+2*t^2+2*t+1)/(630*t^4-35*t^3 - 35*t^2-35*t+1)) \\ _G. C. Greubel_, Dec 11 2016
%o A163220 (Magma) R<x>:=PowerSeriesRing(Integers(), 20); Coefficients(R!( (1+x)*(1-x^4)/(1-36*x+665*x^4-630*x^5) )); // _G. C. Greubel_, Apr 30 2019
%o A163220 (Sage) ((1+x)*(1-x^4)/(1-36*x+665*x^4-630*x^5)).series(x, 20).coefficients(x, sparse=False) # _G. C. Greubel_, Apr 30 2019
%K A163220 nonn
%O A163220 0,2
%A A163220 _John Cannon_ and _N. J. A. Sloane_, Dec 03 2009