A167989 - OEIS

%I #21 Jan 15 2023 02:29:05

%S 1,50,2450,120050,5882450,288240050,14123762450,692064360050,

%T 33911153642450,1661646528480050,81420679895522450,

%U 3989613314880600050,195491052429149402450,9579061569028320720050,469374016882387715282450

%N Number of reduced words of length n in Coxeter group on 50 generators S_i with relations (S_i)^2 = (S_i S_j)^16 = I.

%C The initial terms coincide with those of A170769, 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="/A167989/b167989.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 (48,48,48,48,48,48,48,48,48,48,48,48,48,48,48,-1176).

%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)/( 1176*t^16 - 48*t^15 - 48*t^14 - 48*t^13 - 48*t^12 - 48*t^11 - 48*t^10 - 48*t^9 - 48*t^8 - 48*t^7 - 48*t^6 - 48*t^5 - 48*t^4 - 48*t^3 - 48*t^2 - 48*t + 1).

%F From _G. C. Greubel_, Jan 14 2023: (Start)

%F a(n) = -1176*a(n-16) + 48*Sum_{j=1..15} a(n-j).

%F G.f.: (1+x)*(1-x^16)/(1-49*x+1224*x^16-1176*x^17). (End)

%t CoefficientList[Series[(1+x)*(1-x^16)/(1-49*x+1224*x^16-1176*x^17), {x, 0, 50}], x] (* _G. C. Greubel_, Jul 03 2016; Jan 14 2023 *)

%t coxG[{16, 1176, -48, 10}] (* The coxG program is at A169452 *) (* _G. C. Greubel_, Jan 14 2023 *)

%o (Magma) R<x>:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (1+x)*(1-x^16)/(1-49*x+1224*x^16-1176*x^17) )); // _G. C. Greubel_, Jan 14 2023

%o (SageMath)

%o def A167989_list(prec):

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

%o     return P( (1+x)*(1-x^16)/(1-49*x+1224*x^16-1176*x^17) ).list()

%o A167989_list(40) # _G. C. Greubel_, Jan 14 2023

%Y Cf. A154638, A169452, A170769.

%K nonn

%O 0,2

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