%I #18 Sep 08 2022 08:45:46
%S 1,41,1640,64780,2558400,101024820,3989217180,157523886000,
%T 6220211664420,245620097065980,9698903409405600,382984651654144020,
%U 15123074971766970780,597171180654087109200,23580747941118076783620
%N Number of reduced words of length n in Coxeter group on 41 generators S_i with relations (S_i)^2 = (S_i S_j)^3 = I.
%C The initial terms coincide with those of A170760, although the two sequences are eventually different.
%C Computed with MAGMA using commands similar to those used to compute A154638.
%H G. C. Greubel, <a href="/A162878/b162878.txt">Table of n, a(n) for n = 0..615</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (39, 39, -780).
%F G.f.: (t^3 + 2*t^2 + 2*t + 1)/(780*t^3 - 39*t^2 - 39*t + 1).
%F a(n) = 39*a(n-1) + 39*a(n-2) - 780*a(n-3), n > 0. - _Muniru A Asiru_, Oct 24 2018
%F G.f.: (1+x)*(1-x^3)/(1 - 40*x + 819*x^3 - 780*x^4). - _G. C. Greubel_, Apr 27 2019
%p seq(coeff(series((x^3+2*x^2+2*x+1)/(780*x^3-39*x^2-39*x+1),x,n+1), x, n), n = 0 .. 20); # _Muniru A Asiru_, Oct 24 20182018
%t CoefficientList[Series[(t^3+2*t^2+2*t+1)/(780*t^3-39*t^2-39*t+1), {t, 0, 20}], t] (* _G. C. Greubel_, Oct 24 2018 *)
%t coxG[{3, 780, -39}] (* The coxG program is at A169452 *) (* _G. C. Greubel_, Apr 27 2019 *)
%o (PARI) my(t='t+O('t^20)); Vec((t^3+2*t^2+2*t+1)/(780*t^3-39*t^2-39*t+1)) \\ _G. C. Greubel_, Oct 24 2018
%o (Magma) R<t>:=PowerSeriesRing(Integers(), 20); Coefficients(R!((t^3 + 2*t^2+2*t+1)/(780*t^3-39*t^2-39*t+1))); // _G. C. Greubel_, Oct 24 2018
%o (GAP) a:=[41,1640,64780];; for n in [4..20] do a[n]:=39*a[n-1]+39*a[n-2] -780*a[n-3]; od; Concatenation([1],a); # _Muniru A Asiru_, Oct 24 2018
%o (Sage) ((1+x)*(1-x^3)/(1-40*x+819*x^3-780*x^4)).series(x, 20).coefficients(x, sparse=False) # _G. C. Greubel_, Apr 27 2019
%K nonn
%O 0,2
%A _John Cannon_ and _N. J. A. Sloane_, Dec 03 2009