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%I #19 Sep 08 2022 08:45:48
%S 1,19,342,6156,110808,1994544,35901792,646232256,11632180608,
%T 209379250944,3768826516992,67838877305856,1221099791505408,
%U 21979796247097173,395636332447746036,7121453984059373415
%N Number of reduced words of length n in Coxeter group on 19 generators S_i with relations (S_i)^2 = (S_i S_j)^13 = I.
%C The initial terms coincide with those of A170738, 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="/A167049/b167049.txt">Table of n, a(n) for n = 0..500</a>
%H <a href="/index/Rec#order_13">Index entries for linear recurrences with constant coefficients</a>, signature (17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, -153).
%F G.f.: (t^13 + 2*t^12 + 2*t^11 + 2*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)/(153*t^13 - 17*t^12 - 17*t^11 - 17*t^10 - 17*t^9 - 17*t^8 - 17*t^7 - 17*t^6 - 17*t^5 - 17*t^4 - 17*t^3 - 17*t^2 - 17*t + 1).
%F G.f.: (1+x)*(1-x^13)/(1 - 18*x + 170*x^13 - 153*x^14). - _G. C. Greubel_, Apr 26 2019
%F a(n) = -153*a(n-13) + 17*Sum_{k=1..12} a(n-k). - _Wesley Ivan Hurt_, May 06 2021
%t CoefficientList[Series[(1+x)*(1-x^13)/(1-18*x+170*x^13-153*x^14), {x, 0, 20}], x] (* _G. C. Greubel_, May 31 2016, modified Apr 26 2019 *)
%t coxG[{13, 153, -17}] (* The coxG program is at A169452 *) (* _G. C. Greubel_, Apr 26 2019 *)
%o (PARI) my(x='x+O('x^20)); Vec((1+x)*(1-x^13)/(1-18*x+170*x^13-153*x^14)) \\ _G. C. Greubel_, Apr 26 2019
%o (Magma) R<x>:=PowerSeriesRing(Integers(), 20); Coefficients(R!( (1+x)*(1-x^13)/(1-18*x+170*x^13-153*x^14) )); // _G. C. Greubel_, Apr 26 2019
%o (Sage) ((1+x)*(1-x^13)/(1-18*x+170*x^13-153*x^14)).series(x, 20).coefficients(x, sparse=False) # _G. C. Greubel_, Apr 26 2019
%K nonn
%O 0,2
%A _John Cannon_ and _N. J. A. Sloane_, Dec 03 2009
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