A166420 - OEIS

%I #15 Jul 25 2024 13:58:47

%S 1,26,650,16250,406250,10156250,253906250,6347656250,158691406250,

%T 3967285156250,99182128906250,2479553222655925,61988830566390000,

%U 1549720764159547200,38743019103983610000,968575477599463500000

%N Number of reduced words of length n in Coxeter group on 26 generators S_i with relations (S_i)^2 = (S_i S_j)^11 = I.

%C The initial terms coincide with those of A170745, 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="/A166420/b166420.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 (24,24,24,24,24,24,24,24,24,24,-300).

%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)/(300*t^11 - 24*t^10 - 24*t^9 - 24*t^8 - 24*t^7 - 24*t^6 - 24*t^5 - 24*t^4 - 24*t^3 - 24*t^2 - 24*t + 1).

%F From _G. C. Greubel_, Jan 17 2023: (Start)

%F a(n) = 24*Sum_{j=1..10} a(n-j) - 300*a(n-11).

%F G.f.: (1+x)*(1-x^11)/(1 - 25*x + 324*x^11 - 300*x^12). (End)

%t With[{p=300, q=24}, 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 13 2016; Jul 25 2024 *)

%t coxG[{11,300,-24}] (* The coxG program is at A169452 *) (* _Harvey P. Dale_, Dec 31 2017 *)

%o (Magma)

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

%o Coefficients(R!( (1+x)*(1-x^11)/(1-25*x+324*x^11-300*x^12) )); // _G. C. Greubel_, Jul 25 2024

%o (SageMath)

%o def A166420_list(prec):

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

%o     return P( (1+x)*(1-x^11)/(1-25*x+324*x^11-300*x^12) ).list()

%o A166420_list(30) # _G. C. Greubel_, Jul 25 2024

%Y Cf. A154638, A169452, A170745.

%K nonn

%O 0,2

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