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%I #16 Sep 08 2022 08:45:47
%S 1,29,812,22736,636608,17825024,499100672,13974818816,391294926848,
%T 10956257951338,306775222626096,8589706233212790,240511774521056976,
%U 6734329686340363296,188561231210551675392,5279714473700048997888
%N Number of reduced words of length n in Coxeter group on 29 generators S_i with relations (S_i)^2 = (S_i S_j)^9 = I.
%C The initial terms coincide with those of A170748, 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="/A165512/b165512.txt">Table of n, a(n) for n = 0..689</a>
%H <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (27,27,27,27,27,27,27,27,-378).
%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)/(378*t^9 - 27*t^8 - 27*t^7 - 27*t^6 - 27*t^5 - 27*t^4 - 27*t^3 - 27*t^2 - 27*t + 1).
%p seq(coeff(series((1+t)*(1-t^9)/(1-28*t+405*t^9-378*t^10), t, n+1), t, n), n = 0..20); # _G. C. Greubel_, Sep 16 2019
%t CoefficientList[Series[(1+t)*(1-t^9)/(1-28*t+405*t^9-378*t^10), {t,0, 20}], t] (* _G. C. Greubel_, Oct 21 2018 *)
%t coxG[{9, 378, -27}] (* The coxG program is at A169452 *) (* _G. C. Greubel_, Sep 16 2019 *)
%o (PARI) my(t='t+O('t^20)); Vec((1+t)*(1-t^9)/(1-28*t+405*t^9-378*t^10)) \\ _G. C. Greubel_, Oct 21 2018
%o (Magma) R<t>:=PowerSeriesRing(Integers(), 20); Coefficients(R!( (1+t)*(1-t^9)/(1-28*t+405*t^9-378*t^10) )); // _G. C. Greubel_, Oct 21 2018
%o (Sage)
%o def A165512_list(prec):
%o P.<t> = PowerSeriesRing(ZZ, prec)
%o return P((1+t)*(1-t^9)/(1-28*t+405*t^9-378*t^10)).list()
%o A165512_list(20) # _G. C. Greubel_, Sep 16 2019
%o (GAP) a:=[29, 812, 22736, 636608, 17825024, 499100672, 13974818816, 391294926848, 10956257951338];; for n in [10..20] do a[n]:=27*Sum([1..8], j-> a[n-j]) -378*a[n-9]; od; Concatenation([1], a); # _G. C. Greubel_, Sep 16 2019
%K nonn
%O 0,2
%A _John Cannon_ and _N. J. A. Sloane_, Dec 03 2009
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