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A163677 Number of reduced words of length n in Coxeter group on 41 generators S_i with relations (S_i)^2 = (S_i S_j)^5 = I.

%I A163677 #16 Sep 08 2022 08:45:47
%S A163677 1,41,1640,65600,2624000,104959180,4198334400,167932064820,
%T A163677 6717230145600,268687107936000,10747400402591580,429892659535950000,
%U A163677 17195572119777744420,687817514366631090000,27512485759363357584000
%N A163677 Number of reduced words of length n in Coxeter group on 41 generators S_i with relations (S_i)^2 = (S_i S_j)^5 = I.
%C A163677 The initial terms coincide with those of A170760, although the two sequences are eventually different.
%C A163677 Computed with MAGMA using commands similar to those used to compute A154638.
%H A163677 G. C. Greubel, <a href="/A163677/b163677.txt">Table of n, a(n) for n = 0..620</a>
%H A163677 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (39, 39, 39, 39, -780).
%F A163677 G.f.: (t^5 + 2*t^4 + 2*t^3 + 2*t^2 + 2*t + 1)/(780*t^5 - 39*t^4 - 39*t^3 - 39*t^2 - 39*t + 1).
%F A163677 a(n) = 39*a(n-1)+39*a(n-2)+39*a(n-3)+39*a(n-4)-780*a(n-5). - _Wesley Ivan Hurt_, May 11 2021
%t A163677 CoefficientList[Series[(1+x)*(1-x^5)/(1-40*x+819*x^5-780*x^6), {x,0,20}], x] (* _G. C. Greubel_, Aug 02 2017 *)
%t A163677 coxG[{5, 780, -39}] (* The coxG program is at A169452 *) (* _G. C. Greubel_, May 24 2019 *)
%o A163677 (PARI) my(x='x+O('x^20)); Vec((1+x)*(1-x^5)/(1-40*x+819*x^5-780*x^6)) \\ _G. C. Greubel_, Aug 02 2017
%o A163677 (Magma) R<x>:=PowerSeriesRing(Integers(), 20); Coefficients(R!( (1+x)*(1-x^5)/(1-40*x+819*x^5-780*x^6) )); // _G. C. Greubel_, May 24 2019
%o A163677 (Sage) ((1+x)*(1-x^5)/(1-40*x+819*x^5-780*x^6)).series(x, 20).coefficients(x, sparse=False) # _G. C. Greubel_, May 24 2019
%K A163677 nonn
%O A163677 0,2
%A A163677 _John Cannon_ and _N. J. A. Sloane_, Dec 03 2009