A166583 - OEIS

%I #16 Dec 03 2024 08:20:38

%S 1,15,210,2940,41160,576240,8067360,112943040,1581202560,22136835840,

%T 309915701760,4338819824640,60743477544855,850408685626500,

%U 11905721598750525,166680102382220700,2333521433347076700

%N Number of reduced words of length n in Coxeter group on 15 generators S_i with relations (S_i)^2 = (S_i S_j)^12 = I.

%C The initial terms coincide with those of A170734, 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="/A166583/b166583.txt">Table of n, a(n) for n = 0..500</a>

%H <a href="/index/Rec#order_12">Index entries for linear recurrences with constant coefficients</a>, signature (13,13,13,13,13,13,13,13,13,13,13,-91).

%F G.f.: (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)/(91*t^12 - 13*t^11 - 13*t^10 - 13*t^9 - 13*t^8 - 13*t^7 - 13*t^6 - 13*t^5 - 13*t^4 - 13*t^3 - 13*t^2 - 13*t +1).

%F From _G. C. Greubel_, Dec 03 2024: (Start)

%F a(n) = 13*Sum_{j=1..11} a(n-j) - 91*a(n-12).

%F G.f.: (1+x)*(1-x^12)/(1 - 14*x + 104*x^12 - 91*x^13). (End)

%t CoefficientList[Series[(1+x)*(1-x^12)/(1 - 14*x + 104*x^12 - 91*x^13), {t, 0, 50}], t] (* _G. C. Greubel_, May 17 2016; Dec 03 2024 *)

%t coxG[{12,91,-13}] (* The coxG program is at A169452 *) (* _G. C. Greubel_, Dec 03 2024 *)

%o (Magma)

%o R<x>:=PowerSeriesRing(Integers(), 40);

%o Coefficients(R!( (1+x)*(1-x^12)/(1-14*x+104*x^12-91*x^13) )); // _G. C. Greubel_, Dec 03 2024

%o (SageMath)

%o def A166583_list(prec):

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

%o     return P( (1+x)*(1-x^12)/(1-14*x+104*x^12-91*x^13) ).list()

%o A166583_list(40) # _G. C. Greubel_, Dec 03 2024

%Y Cf. A154638, A169452, A170734.

%K nonn

%O 0,2

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