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%I A164112 #25 Sep 08 2022 08:45:47
%S A164112 1,42,1722,70602,2894682,118681962,4865959581,199504307520,
%T A164112 8179675161840,335366622329760,13750029083987280,563751092750630400,
%U A164112 23113790715369815580,947665251746544828000,38854268450681230932000,1593024724769968897327200,65314002165544342871757600
%N A164112 Number of reduced words of length n in Coxeter group on 42 generators S_i with relations (S_i)^2 = (S_i S_j)^6 = I.
%C A164112 The initial terms coincide with those of A170761, although the two sequences are eventually different.
%C A164112 Computed with MAGMA using commands similar to those used to compute A154638.
%H A164112 G. C. Greubel, <a href="/A164112/b164112.txt">Table of n, a(n) for n = 0..615</a>
%H A164112 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (40,40,40,40,40,-820).
%F A164112 G.f.: (t^6 + 2*t^5 + 2*t^4 + 2*t^3 + 2*t^2 + 2*t + 1)/(820*t^6 - 40*t^5 - 40*t^4 - 40*t^3 - 40*t^2 - 40*t + 1).
%F A164112 a(n) = -820*a(n-6) + 40*Sum_{k=1..5} a(n-k). - _Wesley Ivan Hurt_, May 11 2021
%p A164112 seq(coeff(series((1+t)*(1-t^6)/(1-41*t+860*t^6-820*t^7), t, n+1), t, n), n = 0 .. 30); # _G. C. Greubel_, Aug 16 2019
%t A164112 CoefficientList[Series[(1+t)*(1-t^6)/(1-41*t+860*t^6-820*t^7), {t,0,30}], t] (* _G. C. Greubel_, Sep 11 2017 *)
%t A164112 coxG[{6,820,-40}] (* The coxG program is at A169452 *) (* _Harvey P. Dale_, Jan 16 2018 *)
%o A164112 (PARI) my(t='t+O('t^30)); Vec((1+t)*(1-t^6)/(1-41*t+860*t^6-820*t^7)) \\ _G. C. Greubel_, Sep 11 2017
%o A164112 (Magma) R<t>:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1+t)*(1-t^6)/(1-41*t+860*t^6-820*t^7) )); // _G. C. Greubel_, Aug 16 2019
%o A164112 (Sage)
%o A164112 def A164112_list(prec):
%o A164112     P.<t> = PowerSeriesRing(ZZ, prec)
%o A164112     return P((1+t)*(1-t^6)/(1-41*t+860*t^6-820*t^7)).list()
%o A164112 A164112_list(30) # _G. C. Greubel_, Aug 16 2019
%o A164112 (GAP) a:=[42, 1722, 70602, 2894682, 118681962, 4865959581];; for n in [7..30] do a[n]:=40*(a[n-1] +a[n-2]+a[n-3]+a[n-4]+a[n-5]) -820*a[n-6]; od; Concatenation([1], a); # _G. C. Greubel_, Aug 16 2019
%K A164112 nonn
%O A164112 0,2
%A A164112 _John Cannon_ and _N. J. A. Sloane_, Dec 03 2009

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