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Number of reduced words of length n in Coxeter group on 28 generators S_i with relations (S_i)^2 = (S_i S_j)^10 = I.
1

%I #15 Sep 08 2022 08:45:48

%S 1,28,756,20412,551124,14880348,401769396,10847773692,292889889684,

%T 7908027021468,213516729579258,5764951698629760,155653695862728336,

%U 4202649788286235104,113471544283527738672,3063731695649832497472

%N Number of reduced words of length n in Coxeter group on 28 generators S_i with relations (S_i)^2 = (S_i S_j)^10 = I.

%C The initial terms coincide with those of A170747, 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="/A165980/b165980.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 (26,26,26,26,26,26,26,26,26,-351).

%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)/(351*t^10 - 26*t^9 - 26*t^8 - 26*t^7 - 26*t^6 - 26*t^5 - 26*t^4 - 26*t^3 - 26*t^2 - 26*t + 1).

%p seq(coeff(series((1+t)*(1-t^10)/(1-27*t+377*t^10-351*t^11), t, n+1), t, n), n = 0..30); # _G. C. Greubel_, Oct 25 2019

%t CoefficientList[Series[(1+t)*(1-t^10)/(1-27*t+377*t^10-351*t^11), {t, 0, 30}], t] (* _G. C. Greubel_, Apr 20 2016 *)

%t coxG[{10, 351, -26}] (* The coxG program is at A169452 *) (* _G. C. Greubel_, Oct 25 2019 *)

%o (PARI) my(t='t+O('t^30)); Vec((1+t)*(1-t^10)/(1-27*t+377*t^10-351*t^11)) \\ _G. C. Greubel_, Oct 25 2019

%o (Magma) R<t>:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1+t)*(1-t^10)/(1-27*t+377*t^10-351*t^11) )); // _G. C. Greubel_, Oct 25 2019

%o (Sage)

%o def A165980_list(prec):

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

%o return P((1+t)*(1-t^10)/(1-27*t+377*t^10-351*t^11)).list()

%o A165980_list(30) # _G. C. Greubel_, Oct 25 2019

%o (GAP) a:=[28, 756, 20412, 551124, 14880348, 401769396, 10847773692, 292889889684, 7908027021468, 213516729579258];; for n in [11..30] do a[n]:=26*Sum([1..9], j-> a[n-j]) - 351*a[n-10]; od; Concatenation([1], a); # _G. C. Greubel_, Oct 25 2019

%K nonn

%O 0,2

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