%I #17 Mar 24 2021 08:05:45

%S 1,9,72,576,4608,36864,294912,2359296,18874368,150994944,1207959552,

%T 9663676416,77309411328,618475290624,4947802324992,39582418599936,

%U 316659348799488,2533274790395868,20266198323166656,162129586585330980

%N Number of reduced words of length n in Coxeter group on 9 generators S_i with relations (S_i)^2 = (S_i S_j)^17 = I.

%C The initial terms coincide with those of A003951, although the two sequences are eventually different.

%C First disagreement at index 17: a(17) = 2533274790395868, A003951(17) = 2533274790395904. - _Klaus Brockhaus_, Mar 30 2011

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

%H G. C. Greubel, <a href="/A168686/b168686.txt">Table of n, a(n) for n = 0..500</a>

%H <a href="/index/Rec#order_17">Index entries for linear recurrences with constant coefficients</a>, signature (7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,-28).

%F G.f.: (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)/ (28*t^17 - 7*t^16 - 7*t^15 - 7*t^14 - 7*t^13 - 7*t^12 - 7*t^11 - 7*t^10 - 7*t^9 - 7*t^8 - 7*t^7 - 7*t^6 - 7*t^5 - 7*t^4 - 7*t^3 - 7*t^2 - 7*t + 1).

%F G.f.: (1+t)*(1-t^17)/(1 - 8*t + 35*t^17 - 28*t^18). - _G. C. Greubel_, Mar 24 2021

%t CoefficientList[Series[(1+t)*(1-t^17)/(1 -8*t +35*t^17 -28*t^18), {t, 0, 40}], t] (* _G. C. Greubel_, Aug 03 2016; Mar 24 2021 *)

%t coxG[{17, 28, -7, 40}] (* The coxG program is at A169452 *) (* _G. C. Greubel_, Mar 24 2021 *)

%o (Magma)

%o R<t>:=PowerSeriesRing(Integers(), 40);

%o Coefficients(R!( (1+t)*(1-t^17)/(1 -8*t +35*t^17 -28*t^18) )); // _G. C. Greubel_, Mar 24 2021

%o (Sage)

%o def A168686_list(prec):

%o P.<t> = PowerSeriesRing(ZZ, prec)

%o return P( (1+t)*(1-t^17)/(1 -8*t +35*t^17 -28*t^18) ).list()

%o A168686_list(40) # _G. C. Greubel_, Mar 24 2021

%Y Cf. A003951 (g.f.: (1+x)/(1-8*x)).

%K nonn

%O 0,2

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