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A163923 Number of reduced words of length n in Coxeter group on 7 generators S_i with relations (S_i)^2 = (S_i S_j)^6 = I.

%I A163923 #18 Sep 08 2022 08:45:47
%S A163923 1,7,42,252,1512,9072,54411,326340,1957305,11739420,70410060,
%T A163923 422301600,2532857460,15191434125,91114353750,546480693675,
%U A163923 3277652052150,19658522431800,117906811965600,707175035973000,4241455800274875
%N A163923 Number of reduced words of length n in Coxeter group on 7 generators S_i with relations (S_i)^2 = (S_i S_j)^6 = I.
%C A163923 The initial terms coincide with those of A003949, although the two sequences are eventually different.
%C A163923 Computed with MAGMA using commands similar to those used to compute A154638.
%H A163923 G. C. Greubel, <a href="/A163923/b163923.txt">Table of n, a(n) for n = 0..1000</a>
%H A163923 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (5,5,5,5,5,-15).
%F A163923 G.f.: (t^6 + 2*t^5 + 2*t^4 + 2*t^3 + 2*t^2 + 2*t + 1)/(15*t^6 - 5*t^5 - 5*t^4 - 5*t^3 - 5*t^2 - 5*t + 1).
%F A163923 a(n) = 5*a(n-1)+5*a(n-2)+5*a(n-3)+5*a(n-4)+5*a(n-5)-15*a(n-6). - _Wesley Ivan Hurt_, Apr 23 2021
%p A163923 seq(coeff(series((1+t)*(1-t^6)/(1-6*t+20*t^6-15*t^7), t, n+1), t, n), n = 0 .. 30); # _G. C. Greubel_, Aug 10 2019
%t A163923 coxG[{6,15,-5}] (* The coxG program is at A169452 *) (* _Harvey P. Dale_, Aug 18 2015 *)
%t A163923 CoefficientList[Series[(1+t)*(1-t^6)/(1-6*t+20*t^6-15*t^7), {t,0,30}], t] (* _G. C. Greubel_, Aug 08 2017 *)
%o A163923 (PARI) my(t='t+O('t^30)); Vec((1+t)*(1-t^6)/(1-6*t+20*t^6-15*t^7)) \\ _G. C. Greubel_, Aug 08 2017
%o A163923 (Magma) R<t>:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1+t)*(1-t^6)/(1-6*t+20*t^6-15*t^7) )); // _G. C. Greubel_, Aug 10 2019
%o A163923 (Sage)
%o A163923 def A163923_list(prec):
%o A163923     P.<t> = PowerSeriesRing(ZZ, prec)
%o A163923     return P((1+t)*(1-t^6)/(1-6*t+20*t^6-15*t^7)).list()
%o A163923 A163923_list(30) # _G. C. Greubel_, Aug 10 2019
%o A163923 (GAP) a:=[7,42,252,1512,9072,54411];; for n in [7..30] do a[n]:=5*(a[n-1] +a[n-2]+a[n-3]+a[n-4]+a[n-5]) -15*a[n-6]; od; Concatenation([1], a); # _G. C. Greubel_, Aug 10 2019
%K A163923 nonn
%O A163923 0,2
%A A163923 _John Cannon_ and _N. J. A. Sloane_, Dec 03 2009