A166441 - OEIS

%I #17 Jul 26 2024 05:26:28

%S 1,47,2162,99452,4574792,210440432,9680259872,445291954112,

%T 20483429889152,942237774900992,43342937645445632,1993775131690497991,

%U 91713656057762857860,4218828178657089175245,194066096218225996890780

%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)^11 = 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="/A166441/b166441.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 (45,45,45,45,45,45,45,45,45,45,-1035).

%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)/(1035*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_, Jul 26 2024: (Start)

%F a(n) = 45*Sum_{j=1..10} a(n-j) - 1035*a(n-11).

%F G.f.: (1+x)*(1-x^11)/(1 - 46*x + 1080*x^11 - 1035*x^12). (End)

%t coxG[{11,1035,-45}] (* The coxG program is at A169452 *) (* _Harvey P. Dale_, Jan 16 2015 *)

%t With[{p=1035, q=45}, 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 26 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(1035,45,x) )); // _G. C. Greubel_, Jul 26 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 A166441_list(prec):

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

%o     return P( f(1035,45,x) ).list()

%o A166441_list(30) # _G. C. Greubel_, Jul 26 2024

%Y Cf. A154638, A169452, A170766.

%K nonn

%O 0,2

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