%I #20 Sep 06 2023 17:31:25

%S 1,37,1332,47952,1726272,62145792,2237248512,80540946432,

%T 2899474071552,104381066575872,3757718396731392,135277862282330112,

%U 4870003042163884032,175320109517899825152,6311523942644393705472

%N Number of reduced words of length n in Coxeter group on 37 generators S_i with relations (S_i)^2 = (S_i S_j)^16 = I.

%C The initial terms coincide with those of A170756, 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="/A167953/b167953.txt">Table of n, a(n) for n = 0..500</a>

%H <a href="/index/Rec#order_16">Index entries for linear recurrences with constant coefficients</a>, signature (35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,-630).

%F G.f.: (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)/( 630*t^16 - 35*t^15 - 35*t^14 - 35*t^13 - 35*t^12 - 35*t^11 - 35*t^10 - 35*t^9 - 35*t^8 - 35*t^7 - 35*t^6 - 35*t^5 - 35*t^4 - 35*t^3 - 35*t^2 - 35*t + 1).

%F From _G. C. Greubel_, Sep 06 2023: (Start)

%F G.f.: (1+t)*(1-t^16)/(1 - 36*t + 665*t^16 - 630*t^17).

%F a(n) = 35*Sum_{j=1..15} a(n-j) - 630*a(n-16). (End)

%t CoefficientList[Series[(1+t)*(1-t^16)/(1-36*t+665*t^16-630*t^17), {t, 0, 50}], t] (* _G. C. Greubel_, Jul 02 2016; Sep 06 2023 *)

%t coxG[{16,630,-35}] (* The coxG program is at A169452 *) (* _Harvey P. Dale_, Oct 14 2022 *)

%o (Magma) R<x>:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (1+x)*(1-x^16)/(1-36*x+665*x^16-630*x^17) )); // _G. C. Greubel_, Sep 06 2023

%o (SageMath)

%o def A167955_list(prec):

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

%o return P( (1+x)*(1-x^16)/(1-36*x+665*x^16-630*x^17) ).list()

%o A167955_list(40) # _G. C. Greubel_, Sep 06 2023

%Y Cf. A154638, A169452, A170756.

%K nonn

%O 0,2

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