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A163959 Number of reduced words of length n in Coxeter group on 14 generators S_i with relations (S_i)^2 = (S_i S_j)^6 = I.

%I A163959 #17 Sep 08 2022 08:45:47
%S A163959 1,14,182,2366,30758,399854,5198011,67572960,878433192,11419432752,
%T A163959 148450042104,1929816959616,25087183842630,326127713832648,
%U A163959 4239586491528024,55113665158705608,716465177275138200
%N A163959 Number of reduced words of length n in Coxeter group on 14 generators S_i with relations (S_i)^2 = (S_i S_j)^6 = I.
%C A163959 The initial terms coincide with those of A170733, although the two sequences are eventually different.
%C A163959 Computed with MAGMA using commands similar to those used to compute A154638.
%H A163959 G. C. Greubel, <a href="/A163959/b163959.txt">Table of n, a(n) for n = 0..895</a>
%H A163959 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (12,12,12,12,12,-78).
%F A163959 G.f.: (t^6 + 2*t^5 + 2*t^4 + 2*t^3 + 2*t^2 + 2*t + 1)/(78*t^6 - 12*t^5 - 12*t^4 - 12*t^3 - 12*t^2 - 12*t + 1).
%F A163959 a(n) = -78*a(n-6) + 12*Sum_{k=1..5} a(n-k). - _Wesley Ivan Hurt_, May 11 2021
%p A163959 seq(coeff(series((1+t)*(1-t^6)/(1-13*t+90*t^6-78*t^7), t, n+1), t, n), n = 0 .. 30); # _G. C. Greubel_, Aug 11 2019
%t A163959 CoefficientList[Series[(1+t)*(1-t^6)/(1-13*t+90*t^6-78*t^7), {t,0,30}], t] (* _G. C. Greubel_, Aug 13 2017 *)
%t A163959 coxG[{6, 78, -12}] (* The coxG program is at A169452 *) (* _G. C. Greubel_, Aug 11 2019 *)
%o A163959 (PARI) my(t='t+O('t^30)); Vec((1+t)*(1-t^6)/(1-13*t+90*t^6-78*t^7)) \\ _G. C. Greubel_, Aug 13 2017
%o A163959 (Magma) R<t>:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1+t)*(1-t^6)/(1-13*t+90*t^6-78*t^7) )); // _G. C. Greubel_, Aug 11 2019
%o A163959 (Sage)
%o A163959 def A163959_list(prec):
%o A163959     P.<t> = PowerSeriesRing(ZZ, prec)
%o A163959     return P((1+t)*(1-t^6)/(1-13*t+90*t^6-78*t^7)).list()
%o A163959 A163959_list(30) # _G. C. Greubel_, Aug 11 2019
%o A163959 (GAP) a:=[14, 182, 2366, 30758, 399854, 5198011];; for n in [7..30] do a[n]:=12*(a[n-1] +a[n-2]+a[n-3]+a[n-4]+a[n-5]) -78*a[n-6]; od; Concatenation([1], a); # _G. C. Greubel_, Aug 11 2019
%K A163959 nonn
%O A163959 0,2
%A A163959 _John Cannon_ and _N. J. A. Sloane_, Dec 03 2009