%I #15 Sep 08 2022 08:45:49

%S 1,5,20,80,320,1280,5120,20480,81920,327680,1310720,5242880,20971520,

%T 83886080,335544320,1342177280,5368709120,21474836480,85899345920,

%U 343597383680,1374389534720,5497558138880,21990232555520,87960930222080

%N Number of reduced words of length n in Coxeter group on 5 generators S_i with relations (S_i)^2 = (S_i S_j)^35 = I.

%C The initial terms coincide with those of A003947, 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="/A169546/b169546.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_35">Index entries for linear recurrences with constant coefficients</a>, signature (3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, -6).

%F G.f.: (t^35 + 2*t^34 + 2*t^33 + 2*t^32 + 2*t^31 + 2*t^30 + 2*t^29 + 2*t^28 + 2*t^27 + 2*t^26 + 2*t^25 + 2*t^24 + 2*t^23 + 2*t^22 + 2*t^21 + 2*t^20 + 2*t^19 + 2*t^18 + 2*t^17 + 2*t^16 + 2*t^15 + 2*t^14 + 2*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)/(6*t^35 - 3*t^34 - 3*t^33 - 3*t^32 - 3*t^31 - 3*t^30 - 3*t^29 - 3*t^28 - 3*t^27 - 3*t^26 - 3*t^25 - 3*t^24 - 3*t^23 - 3*t^22 - 3*t^21 - 3*t^20 - 3*t^19 - 3*t^18 - 3*t^17 - 3*t^16 - 3*t^15 - 3*t^14 - 3*t^13 - 3*t^12 - 3*t^11 - 3*t^10 - 3*t^9 - 3*t^8 - 3*t^7 - 3*t^6 - 3*t^5 - 3*t^4 - 3*t^3 - 3*t^2 - 3*t + 1).

%F G.f.: (1+x)*(1-x^35)/(1 - 4*x + 9*x^35 - 6*x^36). - _G. C. Greubel_, Apr 25 2019

%t coxG[{35,6,-3,30}] (* The coxG program is at A169452 *) (* _Harvey P. Dale_, Jan 16 2015 *)

%t CoefficientList[Series[(1+x)*(1-x^35)/(1-4*x+9*x^35-6*x^36), {x, 0, 25}], x] (* _G. C. Greubel_, Apr 25 2019 *)

%o (PARI) my(x='x+O('x^25)); Vec((1+x)*(1-x^35)/(1-4*x+9*x^35-6*x^36)) \\ _G. C. Greubel_, Apr 25 2019

%o (Magma) R<x>:=PowerSeriesRing(Integers(), 25); Coefficients(R!( (1+x)*(1-x^35)/(1-4*x+9*x^35-6*x^36) )); // _G. C. Greubel_, Apr 25 2019

%o (Sage) ((1+x)*(1-x^35)/(1-4*x+9*x^35-6*x^36)).series(x, 25).coefficients(x, sparse=False) # _G. C. Greubel_, Apr 25 2019

%K nonn

%O 0,2

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