A166518 - OEIS

%I #14 Aug 03 2024 02:31:53

%S 1,7,42,252,1512,9072,54432,326592,1959552,11757312,70543872,

%T 423263232,2539579371,15237476100,91424855865,548549130780,

%U 3291294758220,19747768390560,118486609390800,710919650629440,4265517869484480

%N Number of reduced words of length n in Coxeter group on 7 generators S_i with relations (S_i)^2 = (S_i S_j)^12 = I.

%C The initial terms coincide with those of A003949, 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="/A166518/b166518.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 (5,5,5,5,5,5,5,5,5,5,5,-15).

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

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

%F a(n) = 5*Sum_{j=1..11} a(n-j) - 15*a(n-12).

%F G.f.: (1+x)*(1-x^12)/(1 - 6*x + 20*x^12 - 15*x^13). (End)

%t With[{p=15, q=5}, CoefficientList[Series[(1+t)*(1-t^12)/(1 - (q+1)*t + (p+q)*t^12 - p*t^13), {t,0,40}], t]] (* _G. C. Greubel_, May 15 2016; Aug 03 2024 *)

%t coxG[{12, 15, -5, 30}] (* The coxG program is at A169452 *) (* _G. C. Greubel_, Aug 03 2024 *)

%o (PARI) Vec((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)/(15*t^12 - 5*t^11 - 5*t^10 - 5*t^9 - 5*t^8 - 5*t^7 - 5*t^6 - 5*t^5 - 5*t^4 - 5*t^3 - 5*t^2 - 5*t + 1)+O(t^99))

%o (Magma)

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

%o f:= func< p,q,x | (1+x)*(1-x^12)/(1-(q+1)*x+(p+q)*x^12-p*x^13) >;

%o Coefficients(R!( f(15,5,x) )); // _G. C. Greubel_, Aug 03 2024

%o (SageMath)

%o def f(p,q,x): return (1+x)*(1-x^12)/(1-(q+1)*x+(p+q)*x^12-p*x^13)

%o def A166518_list(prec):

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

%o     return P( f(15,5,x) ).list()

%o A166518_list(30) # _G. C. Greubel_, Aug 03 2024

%Y Cf. A003949, A154638, A169452.

%K nonn,easy

%O 0,2

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