A167988 - OEIS

%I #18 Jan 15 2023 02:18:05

%S 1,49,2352,112896,5419008,260112384,12485394432,599298932736,

%T 28766348771328,1380784741023744,66277667569139712,

%U 3181328043318706176,152703746079297896448,7329779811806299029504,351829430966702353416192

%N Number of reduced words of length n in Coxeter group on 49 generators S_i with relations (S_i)^2 = (S_i S_j)^16 = I.

%C The initial terms coincide with those of A170768, 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="/A167988/b167988.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 (47,47,47,47,47,47,47,47,47,47,47,47,47,47,47,-1128).

%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)/( 1128*t^16 - 47*t^15 - 47*t^14 - 47*t^13 - 47*t^12 - 47*t^11 - 47*t^10 - 47*t^9 - 47*t^8 - 47*t^7 - 47*t^6 - 47*t^5 - 47*t^4 - 47*t^3 - 47*t^2 - 47*t + 1).

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

%F a(n) = -1128*a(n-16) + 47*Sum_{j=1..15} a(n-j).

%F G.f.: (1 + x)*(1 - x^16)/(1 - 48*x + 1175*x^16 - 1128*x^17). (End)

%t coxG[{16,1128,-47}] (* The coxG program is at A169452 *) (* _Harvey P. Dale_, May 05 2015 *)

%t CoefficientList[Series[(1+x)*(1-x^16)/(1-48*x+1175*x^16-1128*x^17), {x, 0, 50}], x] (* _G. C. Greubel_, Jul 03 2016; Jan 14 2023 *)

%o (Magma) R<x>:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (1+x)*(1-x^16)/(1-48*x+1175*x^16-1128*x^17) )); // _G. C. Greubel_, Jan 14 2023

%o (Sage)

%o def A167988_list(prec):

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

%o     return P( (1+x)*(1-x^16)/(1-48*x+1175*x^16-1128*x^17) ).list()

%o A167988_list(40) # _G. C. Greubel_, Jan 14 2023

%Y Cf. A154638, A169452, A170768.

%K nonn

%O 0,2

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