A169452 - OEIS

%I #36 Sep 08 2022 08:45:49

%S 1,7,42,252,1512,9072,54432,326592,1959552,11757312,70543872,

%T 423263232,2539579392,15237476352,91424858112,548549148672,

%U 3291294892032,19747769352192,118486616113152,710919696678912,4265518180073472

%N Number of reduced words of length n in Coxeter group on 7 generators S_i with relations (S_i)^2 = (S_i S_j)^33 = I.

%C The initial terms coincide with those of A003949, although the two sequences are eventually different.

%C Computed with MAGMA using commands similar to those used to compute A154638.

%H Alois P. Heinz, <a href="/A169452/b169452.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_33">Index entries for linear recurrences with constant coefficients</a>, signature (5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, -15).

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

%F G.f.: (1+x)*(1-x^33)/(1 - 6*x + 20*x^33 - 15*x^34). - _G. C. Greubel_, May 01 2019

%F a(n) = -15*a(n-33) + 5*Sum_{k=1..32} a(n-k). - _Wesley Ivan Hurt_, May 06 2021

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

%p S:= series(gf,t,101):

%p seq(coeff(S,t,j),j=0..100); # _Robert Israel_, Aug 26 2014

%t coxG[{pwr_,c1_,c2_,trms_:20}]:=Module[{num=Total[2t^Range[pwr-1]]+t^pwr+ 1, den =Total[c2*t^Range[pwr-1]]+c1*t^pwr+1},CoefficientList[ Series[ num/den,{t,0,trms}],t]]; coxG[{33,15,-5,30}]

%t (* "pwr" is the largest exponent in the g.f.;

%t "c1" is the first coefficient in the denominator of the g.f.;

%t "c2" is the second coefficient in the denominator of the g.f.;

%t "trms" is the number of terms desired (with a default number of 20) *)

%t (* _Harvey P. Dale_, Aug 16 2014 *)

%t CoefficientList[Series[(1+x)*(1-x^33)/(1-6*x+20*x^33-15*x^34), {x,0,25}], x] (* _G. C. Greubel_, May 01 2019 *)

%o (PARI) my(x='x+O('x^25)); Vec((1+x)*(1-x^33)/(1-6*x+20*x^33-15*x^34)) \\ _G. C. Greubel_, May 01 2019

%o (Magma) R<x>:=PowerSeriesRing(Integers(), 25); Coefficients(R!( (1+x)*(1-x^3)/(1-14*x+104*x^3-91*x^4) )); // _G. C. Greubel_, May 01 2019

%o (Sage) ((1+x)*(1-x^33)/(1-6*x+20*x^33-15*x^34)).series(x, 25).coefficients(x, sparse=False) # _G. C. Greubel_, May 01 2019

%K nonn,easy

%O 0,2

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