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A163953 Number of reduced words of length n in Coxeter group on 9 generators S_i with relations (S_i)^2 = (S_i S_j)^6 = I.

%I A163953 #18 Sep 08 2022 08:45:47
%S A163953 1,9,72,576,4608,36864,294876,2358720,18867492,150921792,1207229184,
%T A163953 9656672256,77244089580,617878417968,4942433025684,39534710232528,
%U A163953 316239654648960,2529613056079872,20234471292326844,161856307428494112
%N A163953 Number of reduced words of length n in Coxeter group on 9 generators S_i with relations (S_i)^2 = (S_i S_j)^6 = I.
%C A163953 The initial terms coincide with those of A003951, although the two sequences are eventually different.
%C A163953 Computed with MAGMA using commands similar to those used to compute A154638.
%H A163953 G. C. Greubel, <a href="/A163953/b163953.txt">Table of n, a(n) for n = 0..1000</a>
%H A163953 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (7,7,7,7,7,-28).
%F A163953 G.f.: (t^6 + 2*t^5 + 2*t^4 + 2*t^3 + 2*t^2 + 2*t + 1)/(28*t^6 - 7*t^5 - 7*t^4 - 7*t^3 - 7*t^2 - 7*t + 1).
%F A163953 a(n) = -28*a(n-6) + 7*Sum_{k=1..5} a(n-k). - _Wesley Ivan Hurt_, May 11 2021
%p A163953 seq(coeff(series((1+t)*(1-t^6)/(1-8*t+35*t^6-28*t^7), t, n+1), t, n), n = 0 .. 30); # _G. C. Greubel_, Aug 10 2019
%t A163953 CoefficientList[Series[(1+t)*(1-t^6)/(1-8*t+35*t^6-28*t^7), {t, 0, 30}], t] (* _G. C. Greubel_, Aug 13 2017 *)
%t A163953 coxG[{6, 28, -7}] (* The coxG program is at A169452 *) (* _G. C. Greubel_, Aug 10 2019 *)
%o A163953 (PARI) my(t='t+O('t^30)); Vec((1+t)*(1-t^6)/(1-8*t+35*t^6-28*t^7)) \\ _G. C. Greubel_, Aug 13 2017
%o A163953 (Magma) R<t>:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1+t)*(1-t^6)/(1-8*t+35*t^6-28*t^7) )); // _G. C. Greubel_, Aug 10 2019
%o A163953 (Sage)
%o A163953 def A163953_list(prec):
%o A163953     P.<t> = PowerSeriesRing(ZZ, prec)
%o A163953     return P((1+t)*(1-t^6)/(1-8*t+35*t^6-28*t^7)).list()
%o A163953 A163953_list(30) # _G. C. Greubel_, Aug 10 2019
%o A163953 (GAP) a:=[9,72,576,4608,36864,294876];; for n in [7..30] do a[n]:=7*(a[n-1]+a[n-2]+a[n-3]+a[n-4]+a[n-5]) -28*a[n-6]; od; Concatenation([1], a); # _G. C. Greubel_, Aug 10 2019
%K A163953 nonn
%O A163953 0,2
%A A163953 _John Cannon_ and _N. J. A. Sloane_, Dec 03 2009