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A164685 Number of reduced words of length n in Coxeter group on 41 generators S_i with relations (S_i)^2 = (S_i S_j)^7 = I.

%I A164685 #10 Sep 08 2022 08:45:47
%S A164685 1,41,1640,65600,2624000,104960000,4198400000,167935999180,
%T A164685 6717439934400,268697596064820,10747903790145600,429916149507936000,
%U A164685 17196645896401920000,687865832499456000000,27514633165713408671580
%N A164685 Number of reduced words of length n in Coxeter group on 41 generators S_i with relations (S_i)^2 = (S_i S_j)^7 = I.
%C A164685 The initial terms coincide with those of A170760, although the two sequences are eventually different.
%C A164685 Computed with MAGMA using commands similar to those used to compute A154638.
%H A164685 G. C. Greubel, <a href="/A164685/b164685.txt">Table of n, a(n) for n = 0..620</a>
%H A164685 <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (39,39,39,39,39,39,-780).
%F A164685 G.f.: (t^7 + 2*t^6 + 2*t^5 + 2*t^4 + 2*t^3 + 2*t^2 + 2*t + 1)/(780*t^7 - 39*t^6 - 39*t^5 - 39*t^4 - 39*t^3 - 39*t^2 - 39*t + 1).
%p A164685 seq(coeff(series((1+t)*(1-t^7)/(1-40*t+819*t^7-780*t^8), t, n+1), t, n), n = 0..20); # _G. C. Greubel_, Sep 15 2019
%t A164685 CoefficientList[Series[(1+t)*(1-t^7)/(1-40*t+819*t^7-780*t^8), {t, 0, 20}], t] (* _G. C. Greubel_, Sep 15 2019 *)
%t A164685 coxG[{7, 780, -39}] (* The coxG program is at A169452 *) (* _G. C. Greubel_, Sep 15 2019 *)
%o A164685 (PARI) my(t='t+O('t^20)); Vec((1+t)*(1-t^7)/(1-40*t+819*t^7-780*t^8)) \\ _G. C. Greubel_, Sep 15 2019
%o A164685 (Magma) R<t>:=PowerSeriesRing(Integers(), 20); Coefficients(R!( (1+t)*(1-t^7)/(1-40*t+819*t^7-780*t^8) )); // _G. C. Greubel_, Sep 15 2019
%o A164685 (Sage)
%o A164685 def A164685_list(prec):
%o A164685     P.<t> = PowerSeriesRing(ZZ, prec)
%o A164685     return P((1+t)*(1-t^7)/(1-40*t+819*t^7-780*t^8)).list()
%o A164685 A164685_list(20) # _G. C. Greubel_, Sep 15 2019
%o A164685 (GAP) a:=[41,1640,65600,2624000,104960000,4198400000,167935999180];; for n in [8..20] do a[n]:=39*(a[n-1] +a[n-2]+a[n-3]+a[n-4]+a[n-5]+a[n-6]) -780*a[n-7]; od; Concatenation([1], a); # _G. C. Greubel_, Sep 15 2019
%K A164685 nonn
%O A164685 0,2
%A A164685 _John Cannon_ and _N. J. A. Sloane_, Dec 03 2009