%I #21 Sep 08 2022 08:45:49

%S 1,30,870,25230,731670,21218430,615334470,17844699630,517496289270,

%T 15007392388830,435214379276070,12621216999006030,366015292971174870,

%U 10614443496164071230,307818861388758065670,8926746980273983904430

%N Number of reduced words of length n in Coxeter group on 30 generators S_i with relations (S_i)^2 = (S_i S_j)^39 = I.

%C The initial terms coincide with those of A170749, although the two sequences are eventually different.

%C First disagreement is at index 39, the difference is 435. - _Vincenzo Librandi_, Dec 10 2009

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

%H Vincenzo Librandi, <a href="/A170183/b170183.txt">Table of n, a(n) for n = 0..100</a>

%H <a href="/index/Rec#order_39">Index entries for linear recurrences with constant coefficients</a>, signature (28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, -406).

%F G.f.: (t^39 + 2*t^38 + 2*t^37 + 2*t^36 + 2*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) /(406*t^39 - 28*t^38 - 28*t^37 - 28*t^36 - 28*t^35 - 28*t^34 - 28*t^33 - 28*t^32 - 28*t^31 - 28*t^30 - 28*t^29 - 28*t^28 - 28*t^27 - 28*t^26 - 28*t^25 - 28*t^24 - 28*t^23 - 28*t^22 - 28*t^21 - 28*t^20 - 28*t^19 - 28*t^18 - 28*t^17 - 28*t^16 - 28*t^15 - 28*t^14 - 28*t^13 - 28*t^12 - 28*t^11 - 28*t^10 - 28*t^9 - 28*t^8 - 28*t^7 - 28*t^6 - 28*t^5 - 28*t^4 - 28*t^3 - 28*t^2 - 28*t + 1)

%t With[{num=Total[2t^Range[38]]+t^39+1,den=Total[-28 t^Range[38]]+ 406t^39+1}, CoefficientList[Series[num/den,{t,0,30}],t]] (* _Harvey P. Dale_, Sep 20 2011 *)

%o (Magma) /* Alternatively */ m:=16; R<t>:=PowerSeriesRing(Integers(), m); Coefficients(R!((t^40+t^39-t-1)/(406*t^40-434*t^39+29*t-1))); // _Bruno Berselli_, Sep 20 2011

%Y Cf. A170749.

%K nonn

%O 0,2

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