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A163922 Number of reduced words of length n in Coxeter group on 6 generators S_i with relations (S_i)^2 = (S_i S_j)^6 = I.

%I A163922 #21 Sep 08 2022 08:45:47
%S A163922 1,6,30,150,750,3750,18735,93600,467640,2336400,11673000,58320000,
%T A163922 291375210,1455753000,7273154040,36337737000,181548627000,
%U A163922 907043385000,4531720872060,22641137570400,113118421225440,565156109349600
%N A163922 Number of reduced words of length n in Coxeter group on 6 generators S_i with relations (S_i)^2 = (S_i S_j)^6 = I.
%C A163922 The initial terms coincide with those of A003948, although the two sequences are eventually different.
%C A163922 Computed with MAGMA using commands similar to those used to compute A154638.
%H A163922 G. C. Greubel, <a href="/A163922/b163922.txt">Table of n, a(n) for n = 0..1000</a>
%H A163922 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (4,4,4,4,4,-10).
%F A163922 G.f.: (t^6 + 2*t^5 + 2*t^4 + 2*t^3 + 2*t^2 + 2*t + 1)/(10*t^6 - 4*t^5 - 4*t^4 - 4*t^3 - 4*t^2 - 4*t + 1).
%F A163922 a(n) = -10*a(n-6) + 4*Sum_{k=1..5} a(n-k). - _Wesley Ivan Hurt_, May 11 2021
%p A163922 seq(coeff(series((1+t)*(1-t^6)/(1-5*t+14*t^6-10*t^7), t, n+1), t, n), n = 0 .. 30); # _G. C. Greubel_, Aug 10 2019
%t A163922 coxG[{6,10,-4,30}] (* The coxG program is at A169452 *) (* _Harvey P. Dale_, May 13 2017 *)
%t A163922 CoefficientList[Series[(1+t)*(1-t^6)/(1-5*t+14*t^6-10*t^7), {t,0,30}], t] (* _G. C. Greubel_, Aug 07 2017 *)
%o A163922 (PARI) my(t='t+O('t^30)); Vec((1+t)*(1-t^6)/(1-5*t+14*t^6-10*t^7)) \\ _G. C. Greubel_, Aug 07 2017
%o A163922 (Magma) R<t>:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1+t)*(1-t^6)/(1-5*t+14*t^6-10*t^7) )); // _G. C. Greubel_, Aug 10 2019
%o A163922 (Sage)
%o A163922 def A163922_list(prec):
%o A163922     P.<t> = PowerSeriesRing(ZZ, prec)
%o A163922     return P((1+t)*(1-t^6)/(1-5*t+14*t^6-10*t^7)).list()
%o A163922 A163922_list(30) # _G. C. Greubel_, Aug 10 2019
%o A163922 (GAP) a:=[6,30,150,750,3750,18735];; for n in [7..30] do a[n]:=4*(a[n-1] +a[n-2]+a[n-3]+a[n-4]+a[n-5]) -10*a[n-6]; od; Concatenation([1], a); # _G. C. Greubel_, Aug 10 2019
%K A163922 nonn
%O A163922 0,2
%A A163922 _John Cannon_ and _N. J. A. Sloane_, Dec 03 2009