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Number of reduced words of length n in Coxeter group on 6 generators S_i with relations (S_i)^2 = (S_i S_j)^5 = I.
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%I #17 Sep 08 2022 08:45:46

%S 1,6,30,150,750,3735,18600,92640,461400,2298000,11445210,57003000,

%T 283904040,1413987000,7042377000,35074632060,174689570400,

%U 870043225440,4333259349600,21581843340000,107488595621160,535348070440800

%N Number of reduced words of length n in Coxeter group on 6 generators S_i with relations (S_i)^2 = (S_i S_j)^5 = I.

%C The initial terms coincide with those of A003948, 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="/A163317/b163317.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (4, 4, 4, 4, -10).

%F G.f.: (t^5 + 2*t^4 + 2*t^3 + 2*t^2 + 2*t + 1)/(10*t^5 - 4*t^4 - 4*t^3 - 4*t^2 - 4*t + 1).

%F a(n) = 4*a(n-1)+4*a(n-2)+4*a(n-3)+4*a(n-4)-10*a(n-5). - _Wesley Ivan Hurt_, May 10 2021

%t CoefficientList[Series[(1+x)*(1-x^5)/(1-5*x+14*x^5-10*x^6), {x, 0, 30}], x] (* or *) LinearRecurrence[{4,4,4,4,-10}, {1,6,30,150,750,3735}, 30] (* _G. C. Greubel_, Dec 18 2016 *)

%t coxG[{5, 10, -4}] (* The coxG program is at A169452 *) (* _G. C. Greubel_, May 12 2019 *)

%o (PARI) my(x='x+O('x^30)); Vec((1+x)*(1-x^5)/(1-5*x+14*x^5-10*x^6)) \\ _G. C. Greubel_, Dec 18 2016

%o (Magma) R<x>:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1+x)*(1-x^5)/(1-5*x+14*x^5-10*x^6) )); // _G. C. Greubel_, May 12 2019

%o (Sage) ((1+x)*(1-x^5)/(1-5*x+14*x^5-10*x^6)).series(x, 30).coefficients(x, sparse=False) # _G. C. Greubel_, May 12 2019

%K nonn

%O 0,2

%A _John Cannon_ and _N. J. A. Sloane_, Dec 03 2009