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%I #22 Sep 08 2022 08:45:47
%S 1,27,702,18252,474552,12338352,320797152,8340725952,216858874752,
%T 5638330743201,146596599314100,3811511581929675,99099301124011500,
%U 2576581829064137700,66991127551503386400,1741769316230819007600
%N Number of reduced words of length n in Coxeter group on 27 generators S_i with relations (S_i)^2 = (S_i S_j)^9 = I.
%C The initial terms coincide with those of A170746, 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="/A165445/b165445.txt">Table of n, a(n) for n = 0..700</a>
%H <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (25,25,25,25,25,25,25,25,-325).
%F G.f.: (t^9 + 2*t^8 + 2*t^7 + 2*t^6 + 2*t^5 + 2*t^4 + 2*t^3 + 2*t^2 + 2*t + 1)/(325*t^9 - 25*t^8 - 25*t^7 - 25*t^6 - 25*t^5 - 25*t^4 - 25*t^3 - 25*t^2 - 25*t + 1).
%p seq(coeff(series((x^9+2*x^8+2*x^7+2*x^6+2*x^5+2*x^4+2*x^3+2*x^2+2*x+1 )/(325*x^9-25*x^8-25*x^7-25*x^6-25*x^5-25*x^4-25*x^3-25*x^2 -25*x +1),x,n+1), x, n), n = 0 .. 15); # _Muniru A Asiru_, Oct 21 2018
%t coxG[{9,325,-25}] (* The coxG program is at A169452 *) (* _Harvey P. Dale_, Sep 21 2017 *)
%t CoefficientList[Series[(1+t)*(1-t^9)/(1-26*t+350*t^9-325*t^10), {t, 0, 20}], t] (* _G. C. Greubel_, Oct 20 2018 *)
%o (PARI) t='t+O('t^20); Vec((1+t)*(1-t^9)/(1-26*t+350*t^9-325*t^10)) \\ _G. C. Greubel_, Oct 20 2018
%o (Magma) R<t>:=PowerSeriesRing(Integers(), 20); Coefficients(R!( (1+t)*(1-t^9)/(1-26*t+350*t^9-325*t^10) )); // _G. C. Greubel_, Oct 20 2018
%o (Sage)
%o def A165445_list(prec):
%o P.<t> = PowerSeriesRing(ZZ, prec)
%o return P((1+t)*(1-t^9)/(1-26*t+350*t^9-325*t^10)).list()
%o A165445_list(20) # _G. C. Greubel_, Sep 16 2019
%o (GAP) a:=[27, 702, 18252, 474552, 12338352, 320797152, 8340725952, 216858874752, 5638330743201];; for n in [10..20] do a[n]:=25*Sum([1..8], j-> a[n-j]) -325*a[n-9]; od; Concatenation([1], a); # _G. C. Greubel_, Sep 16 2019
%K nonn,easy
%O 0,2
%A _John Cannon_ and _N. J. A. Sloane_, Dec 03 2009
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