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%I #19 Sep 08 2022 08:45:48
%S 1,20,380,7220,137180,2606420,49521980,940917620,17877434780,
%T 339671260820,6453753955390,122621325148800,2329805177758800,
%U 44266298376117600,841059669121542000,15980133712840142400
%N Number of reduced words of length n in Coxeter group on 20 generators S_i with relations (S_i)^2 = (S_i S_j)^10 = I.
%C The initial terms coincide with those of A170739, 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="/A165882/b165882.txt">Table of n, a(n) for n = 0..500</a>
%H <a href="/index/Rec#order_10">Index entries for linear recurrences with constant coefficients</a>, signature (18,18,18,18,18,18,18,18,18,-171).
%F G.f.: (t^10 + 2*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)/(171*t^10 - 18*t^9 - 18*t^8 - 18*t^7 - 18*t^6 - 18*t^5 - 18*t^4 - 18*t^3 - 18*t^2 - 18*t + 1).
%p seq(coeff(series((1+t)*(1-t^10)/(1-19*t+189*t^10-171*t^11), t, n+1), t, n), n = 0..20); # _G. C. Greubel_, Sep 24 2019
%t CoefficientList[Series[(1+t)*(1-t^10)/(1-19*t+189*t^10-171*t^11), {t, 0, 20}], t] (* _G. C. Greubel_, Apr 17 2016 *)
%t coxG[{10,171,-18}] (* The coxG program is at A169452 *) (* _Harvey P. Dale_, Dec 18 2018 *)
%o (PARI) my(t='t+O('t^20)); Vec((1+t)*(1-t^10)/(1-19*t+189*t^10-171*t^11)) \\ _G. C. Greubel_, Sep 24 2019
%o (Magma) R<t>:=PowerSeriesRing(Integers(), 20); Coefficients(R!( (1+t)*(1-t^10)/(1-19*t+189*t^10-171*t^11) )); // _G. C. Greubel_, Sep 24 2019
%o (Sage)
%o def A165882_list(prec):
%o P.<t> = PowerSeriesRing(ZZ, prec)
%o return P((1+t)*(1-t^10)/(1-19*t+189*t^10-171*t^11)).list()
%o A165882_list(20) # _G. C. Greubel_, Sep 24 2019
%o (GAP) a:=[20, 380, 7220, 137180, 2606420, 49521980, 940917620, 17877434780, 339671260820, 6453753955390];; for n in [11..20] do a[n]:=18*Sum([1..9], j-> a[n-j]) -171*a[n-10]; od; Concatenation([1], a); # _G. C. Greubel_, Sep 24 2019
%Y Cf. A154638, A170739.
%K nonn
%O 0,2
%A _John Cannon_ and _N. J. A. Sloane_, Dec 03 2009
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