A165757 - OEIS

%I #19 Sep 08 2022 08:45:48

%S 1,5,20,80,320,1280,5120,20480,81920,327680,1310710,5242800,20971050,

%T 83883600,335532000,1342118400,5368435200,21473587200,85893734400,

%U 343572480000,1374280089690,5497081037820,21988166868630,87952038348420

%N Number of reduced words of length n in Coxeter group on 5 generators S_i with relations (S_i)^2 = (S_i S_j)^10 = I.

%C The initial terms coincide with those of A003947, 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="/A165757/b165757.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 (3,3,3,3,3,3,3,3,3,-6).

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

%p seq(coeff(series((1+t)*(1-t^10)/(1-4*t+9*t^10-6*t^11), t, n+1), t, n), n = 0..30); # _G. C. Greubel_, Sep 17 2019

%t CoefficientList[Series[(1+t)*(1-t^10)/(1-4*t+9*t^10-6*t^11), {t, 0, 30}], t] (* _G. C. Greubel_, Apr 07 2016 *)

%t coxG[{10, 6, -3}] (* The coxG program is at A169452 *) (* _G. C. Greubel_, Sep 17 2019 *)

%o (PARI) my(t='t+O('t^30)); Vec((1+t)*(1-t^10)/(1-4*t+9*t^10-6*t^11)) \\ _G. C. Greubel_, Sep 17 2019

%o (Magma) R<t>:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1+t)*(1-t^10)/(1-4*t+9*t^10-6*t^11) )); // _G. C. Greubel_, Sep 17 2019

%o (Sage)

%o def A165757_list(prec):

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

%o     return P((1+t)*(1-t^10)/(1-4*t+9*t^10-6*t^11)).list()

%o A165757_list(30) # _G. C. Greubel_, Sep 17 2019

%o (GAP) a:=[5, 20, 80, 320, 1280, 5120, 20480, 81920, 327680, 1310710];; for n in [11..30] do a[n]:=3*Sum([1..9], j-> a[n-j]) -6*a[n-10]; od; Concatenation([1], a); # _G. C. Greubel_, Sep 17 2019

%K nonn

%O 0,2

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