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A165807 Number of reduced words of length n in Coxeter group on 12 generators S_i with relations (S_i)^2 = (S_i S_j)^10 = I. 1

%I #23 Sep 08 2022 08:45:48

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

%T 28295372226,311249093760,3423740023440,37661140170720,

%U 414272540919600,4556997939574080,50126977219358160,551396748137415840

%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)^10 = I.

%C The initial terms coincide with those of A003954, although the two sequences are eventually different.

%C First disagreement at index 10: a(10) = 28295372226, A003954(10) = 28295372292. - _Klaus Brockhaus_, Jun 14 2011

%C Computed with MAGMA using commands similar to those used to compute A154638.

%H G. C. Greubel, <a href="/A165807/b165807.txt">Table of n, a(n) for n = 0..500</a>

%H <a href="/index/Rec#order_10">Index entries for linear recurrences with constant coefficients</a>, signature (10,10,10,10,10,10,10,10,10,-55).

%F G.f.: (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^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).

%p seq(coeff(series((1+t)*(1-t^10)/(1-11*t+65*t^10-55*t^11), t, n+1), t, n), n = 0..20); # _G. C. Greubel_, Sep 23 2019

%t With[{num=Total[2t^Range[9]]+1+t^10,den=Total[-10 t^Range[9]]+1+ 55t^10}, CoefficientList[Series[num/den,{t,0,30}],t]] (* _Harvey P. Dale_, Jun 14 2011 *)

%t CoefficientList[Series[(1+t)*(1-t^10)/(1-11*t+65*t^10-55*t^11), {t, 0, 30}], t] (* or *) coxG[{10, 55, -10}] (* The coxG program is at A169452 *) (* _G. C. Greubel_, Sep 23 2019 *)

%o (PARI) my(t='t+O('t^20)); Vec((1+t)*(1-t^10)/(1-11*t+65*t^10-55*t^11)) \\ _G. C. Greubel_, Sep 23 2019

%o (Magma) R<t>:=PowerSeriesRing(Integers(), 20); Coefficients(R!( (1+t)*(1-t^10)/(1-11*t+65*t^10-55*t^11) )); // _G. C. Greubel_, Sep 23 2019

%o (Sage)

%o def A165807_list(prec):

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

%o return P((1+t)*(1-t^10)/(1-11*t+65*t^10-55*t^11)).list()

%o A165807_list(20) # _G. C. Greubel_, Sep 23 2019

%o (GAP) a:=[12, 132, 1452, 15972, 175692, 1932612, 21258732, 233846052, 2572306572, 28295372226];; for n in [11..20] do a[n]:=10*Sum([1..9], j-> a[n-j]) -55*a[n-10]; od; Concatenation([1], a); # _G. C. Greubel_, Sep 23 2019

%Y Cf. A003954 (G.f.: (1+x)/(1-11*x)).

%K nonn

%O 0,2

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

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Last modified March 28 20:05 EDT 2024. Contains 371254 sequences. (Running on oeis4.)