|
%I #15 Jan 17 2023 07:18:52
%S 1,47,2162,99452,4574792,210440432,9680259872,445291954112,
%T 20483429889152,942237774900992,43342937645445632,1993775131690499072,
%U 91713656057762957312,4218828178657096036352,194066096218226417672192
%N Number of reduced words of length n in Coxeter group on 47 generators S_i with relations (S_i)^2 = (S_i S_j)^16 = I.
%C The initial terms coincide with those of A170766, 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="/A167978/b167978.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 (45,45,45,45,45,45,45,45,45,45,45,45,45,45,45,-1035).
%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)/( 1035*t^16 - 45*t^15 - 45*t^14 - 45*t^13 - 45*t^12 - 45*t^11 - 45*t^10 - 45*t^9 - 45*t^8 - 45*t^7 - 45*t^6 - 45*t^5 - 45*t^4 - 45*t^3 - 45*t^2 - 45*t + 1).
%F From _G. C. Greubel_, Jan 17 2023: (Start)
%F a(n) = Sum_{j=1..15} a(n-j) - 1035*a(n-16).
%F G.f.: (1+x)*(1-x^16)/(1 - 46*x + 1081*x^16 - 1035*x^17). (End)
%t CoefficientList[Series[(1+t)*(1-t^16)/(1-46*t+1081*t^16-1035*t^17), {t, 0,50}], t] (* _G. C. Greubel_, Jul 03 2016; Jan 17 2023 *)
%t coxG[{16,1035,-45}] (* The coxG program is at A169452 *) (* _G. C. Greubel_, Jan 17 2023 *)
%o (Magma) R<x>:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1+x)*(1-x^17)/(1-46*x+1081*x^16-1035*x^17) )); // _G. C. Greubel_, Jan 17 2023
%o (SageMath)
%o def A167978_list(prec):
%o P.<x> = PowerSeriesRing(ZZ, prec)
%o return P( (1+x)*(1-x^17)/(1-46*x+1081*x^16-1035*x^17) ).list()
%o A167978_list(30) # _G. C. Greubel_, Jan 17 2023
%Y Cf. A154638, A169452, A170766.
%K nonn
%O 0,2
%A _John Cannon_ and _N. J. A. Sloane_, Dec 03 2009
| 