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Number of reduced words of length n in Coxeter group on 12 generators S_i with relations (S_i)^2 = (S_i S_j)^12 = I.
1

%I #13 Dec 03 2024 03:24:45

%S 1,12,132,1452,15972,175692,1932612,21258732,233846052,2572306572,

%T 28295372292,311249095212,3423740047266,37661140519200,

%U 414272545703280,4556998002648960,50126978028180240,551396758299441120,6065364341177895600,66719007751681327680

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

%C The initial terms coincide with those of A003954, 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="/A166557/b166557.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 (10,10,10,10,10,10,10,10,10,10,10,-55).

%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)/(55*t^12 - 10*t^11 - 10*t^10 - 10*t^9 - 10*t^8 - 10*t^7 - 10*t^6 - 10*t^5 - 10*t^4 - 10*t^3 - 10*t^2 - 10*t +1).

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

%F a(n) = 10*Sum_{j=1..11} a(n-j) - 55*a(n-12).

%F G.f.: (1+t)*(1 - t^12)/(1 - 11*t + 65*t^12 - 55*t^13). (End)

%t CoefficientList[Series[(1+t)*(1-t^12)/(1-11*t+65*t^12-55*t^13), {t,0,50}], t]

%t (* _G. C. Greubel_, May 17 2016; Dec 03 2024 *)

%t coxG[{12,55,-10}] (* 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)/(1-11*x+65*x^12-55*x^13) )); // ~~~

%o (SageMath)

%o def A166557_list(prec):

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

%o return P( (1+x)*(1-x^12)/(1-11*x+65*x^12-55*x^13) ).list()

%o A166557_list(40) # _G. C. Greubel_, Dec 03 2024

%Y Cf. A003954, A154638, A169452.

%K nonn

%O 0,2

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