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A163967 Number of reduced words of length n in Coxeter group on 18 generators S_i with relations (S_i)^2 = (S_i S_j)^6 = I.

%I A163967 #16 Sep 08 2022 08:45:47
%S A163967 1,18,306,5202,88434,1503378,25557273,434471040,7385963616,
%T A163967 125560632384,2134518016032,36286589786112,616868346117816,
%U A163967 10486699320193920,178272824864888448,3030619941978033024,51520231643134128768
%N A163967 Number of reduced words of length n in Coxeter group on 18 generators S_i with relations (S_i)^2 = (S_i S_j)^6 = I.
%C A163967 The initial terms coincide with those of A170737, although the two sequences are eventually different.
%C A163967 Computed with MAGMA using commands similar to those used to compute A154638.
%H A163967 G. C. Greubel, <a href="/A163967/b163967.txt">Table of n, a(n) for n = 0..800</a>
%H A163967 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (16,16,16,16,16,-136).
%F A163967 G.f.: (t^6 + 2*t^5 + 2*t^4 + 2*t^3 + 2*t^2 + 2*t + 1)/(136*t^6 - 16*t^5 - 16*t^4 - 16*t^3 - 16*t^2 - 16*t + 1).
%F A163967 a(n) = -136*a(n-6) + 16*Sum_{k=1..5} a(n-k). - _Wesley Ivan Hurt_, May 11 2021
%p A163967 seq(coeff(series((1+t)*(1-t^6)/(1-17*t+152*t^6-136*t^7), t, n+1), t, n), n = 0 .. 30); # _G. C. Greubel_, Aug 11 2019
%t A163967 CoefficientList[Series[(1+t)*(1-t^6)/(1-17*t+152*t^6-136*t^7), {t,0,30}], t] (* _G. C. Greubel_, Aug 23 2017 *)
%t A163967 coxG[{6, 136, -16}] (* The coxG program is at A169452 *) (* _G. C. Greubel_, Aug 11 2019 *)
%o A163967 (PARI) my(t='t+O('t^30)); Vec((1+t)*(1-t^6)/(1-17*t+152*t^6-136*t^7)) \\ _G. C. Greubel_, Aug 23 2017
%o A163967 (Magma) R<t>:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1+t)*(1-t^6)/(1-17*t+152*t^6-136*t^7) )); // _G. C. Greubel_, Aug 11 2019
%o A163967 (Sage)
%o A163967 def A163967_list(prec):
%o A163967     P.<t> = PowerSeriesRing(ZZ, prec)
%o A163967     return P((1+t)*(1-t^6)/(1-17*t+152*t^6-136*t^7)).list()
%o A163967 A163967_list(30) # _G. C. Greubel_, Aug 11 2019
%o A163967 (GAP) a:=[18, 306, 5202, 88434, 1503378, 25557273];; for n in [7..30] do a[n]:=16*(a[n-1] +a[n-2]+a[n-3]+a[n-4]+a[n-5]) -136*a[n-6]; od; Concatenation([1], a); # _G. C. Greubel_, Aug 11 2019
%K A163967 nonn
%O A163967 0,2
%A A163967 _John Cannon_ and _N. J. A. Sloane_, Dec 03 2009