A165874 - OEIS

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

%S 1,14,182,2366,30758,399854,5198102,67575326,878479238,11420230094,

%T 148462991131,1930018883520,25090245470472,326173190917392,

%U 4240251479342424,55123269197863776,716602499135588520

%N Number of reduced words of length n in Coxeter group on 14 generators S_i with relations (S_i)^2 = (S_i S_j)^10 = I.

%C The initial terms coincide with those of A170733, 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="/A165874/b165874.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 (12,12,12,12,12,12,12,12,12,-78).

%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)/(78*t^10 - 12*t^9 - 12*t^8 - 12*t^7 - 12*t^6 - 12*t^5 - 12*t^4 - 12*t^3 - 12*t^2 - 12*t + 1).

%p seq(coeff(series((1+t)*(1-t^10)/(1-13*t+90*t^10-78*t^11), t, n+1), t, n), n = 0..20); # _G. C. Greubel_, Sep 23 2019

%t CoefficientList[Series[(1+t)*(1-t^10)/(1-13*t+90*t^10-78*t^11), {t, 0, 25}], t] (* _G. C. Greubel_, Apr 17 2016 *)

%t coxG[{10, 78, -12}] (* 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)/(1-13*t+90*t^10-78*t^11)) \\ _G. C. Greubel_, Sep 23 2019

%o (Magma) R<t>:=PowerSeriesRing(Integers(), 20); Coefficients(R!( (1+t)*(1-t^10)/(1-13*t+90*t^10-78*t^11) )); // _G. C. Greubel_, Sep 23 2019

%o (Sage)

%o def A165874_list(prec):

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

%o     return P((1+t)*(1-t^10)/(1-13*t+90*t^10-78*t^11)).list()

%o A165874_list(20) # _G. C. Greubel_, Sep 23 2019

%o (GAP) a:=[14, 182, 2366, 30758, 399854, 5198102, 67575326, 878479238, 11420230094, 148462991131];; for n in [11..20] do a[n]:=12*Sum([1..9], j-> a[n-j]) -78*a[n-10]; od; Concatenation([1], a); # _G. C. Greubel_, Sep 23 2019

%K nonn

%O 0,2

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