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A169452 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.

%I A169452 #36 Sep 08 2022 08:45:49
%S A169452 1,7,42,252,1512,9072,54432,326592,1959552,11757312,70543872,
%T A169452 423263232,2539579392,15237476352,91424858112,548549148672,
%U A169452 3291294892032,19747769352192,118486616113152,710919696678912,4265518180073472
%N A169452 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 A169452 The initial terms coincide with those of A003949, although the two sequences are eventually different.
%C A169452 Computed with MAGMA using commands similar to those used to compute A154638.
%H A169452 Alois P. Heinz, <a href="/A169452/b169452.txt">Table of n, a(n) for n = 0..1000</a>
%H A169452 <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 A169452 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 A169452 G.f.: (1+x)*(1-x^33)/(1 - 6*x + 20*x^33 - 15*x^34). - _G. C. Greubel_, May 01 2019
%F A169452 a(n) = -15*a(n-33) + 5*Sum_{k=1..32} a(n-k). - _Wesley Ivan Hurt_, May 06 2021
%p A169452 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 A169452 S:= series(gf,t,101):
%p A169452 seq(coeff(S,t,j),j=0..100); # _Robert Israel_, Aug 26 2014
%t A169452 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 A169452 (* "pwr" is the largest exponent in the g.f.;
%t A169452 "c1" is the first coefficient in the denominator of the g.f.;
%t A169452 "c2" is the second coefficient in the denominator of the g.f.;
%t A169452 "trms" is the number of terms desired (with a default number of 20) *)
%t A169452 (* _Harvey P. Dale_, Aug 16 2014 *)
%t A169452 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 A169452 (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 A169452 (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 A169452 (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 A169452 nonn,easy
%O A169452 0,2
%A A169452 _John Cannon_ and _N. J. A. Sloane_, Dec 03 2009