A166443 - OEIS

%I #17 Jul 27 2024 09:40:54

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

%T 28766348771328,1380784741023744,66277667569139712,

%U 3181328043318705000,152703746079297783552,7329779811806290902168,351829430966701833304320

%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)^11 = 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="/A166443/b166443.txt">Table of n, a(n) for n = 0..500</a>

%H <a href="/index/Rec#order_11">Index entries for linear recurrences with constant coefficients</a>, signature (47,47,47,47,47,47,47,47,47,47,-1128).

%F G.f.: (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^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_, Jul 27 2024: (Start)

%F a(n) = 47*Sum_{j=1..10} a(n-j) - 1128*a(n-11).

%F G.f.: (1+x)*(1-x^11)/(1 - 48*x + 1175*x^11 - 1128*x^12). (End)

%t With[{num=Total[2t^Range[10]]+t^11+1,den=Total[-47 t^Range[10]]+ 1128t^11+ 1}, CoefficientList[Series[num/den,{t,0,20}],t]] (* _Harvey P. Dale_, Aug 29 2011 *)

%t With[{p=1128, q=47}, CoefficientList[Series[(1+t)*(1-t^11)/(1-(q+1)*t + (p+q)*t^11 - p*t^12), {t,0,40}], t]] (* _G. C. Greubel_, May 14 2016; Jul 27 2024 *)

%t coxG[{11, 1128, -47, 30}] (* The coxG program is at A169452 *) (* _G. C. Greubel_, Jul 27 2024 *)

%o (Magma)

%o R<x>:=PowerSeriesRing(Integers(), 30);

%o f:= func< p,q,x | (1+x)*(1-x^11)/(1-(q+1)*x+(p+q)*x^11-p*x^12) >;

%o Coefficients(R!( f(1128,47,x) )); // _G. C. Greubel_, Jul 27 2024

%o (SageMath)

%o def f(p,q,x): return (1+x)*(1-x^11)/(1-(q+1)*x+(p+q)*x^11-p*x^12)

%o def A166443_list(prec):

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

%o     return P( f(1128,47,x) ).list()

%o A166443_list(30) # _G. C. Greubel_, Jul 27 2024

%Y Cf. A154638, A169452, A170768.

%K nonn

%O 0,2

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