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A163964 Number of reduced words of length n in Coxeter group on 16 generators S_i with relations (S_i)^2 = (S_i S_j)^6 = I.

%I A163964 #16 Sep 08 2022 08:45:47
%S A163964 1,16,240,3600,54000,810000,12149880,182246400,2733669120,41004633600,
%T A163964 615063456000,9225861120000,138386556014280,2075777928630000,
%U A163964 31136362758696720,467040848864310000,7005543845899914000
%N A163964 Number of reduced words of length n in Coxeter group on 16 generators S_i with relations (S_i)^2 = (S_i S_j)^6 = I.
%C A163964 The initial terms coincide with those of A170735, although the two sequences are eventually different.
%C A163964 Computed with MAGMA using commands similar to those used to compute A154638.
%H A163964 G. C. Greubel, <a href="/A163964/b163964.txt">Table of n, a(n) for n = 0..845</a>
%H A163964 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (14,14,14,14,14,-105).
%F A163964 G.f.: (t^6 + 2*t^5 + 2*t^4 + 2*t^3 + 2*t^2 + 2*t + 1)/(105*t^6 - 14*t^5 - 14*t^4 - 14*t^3 - 14*t^2 - 14*t + 1).
%F A163964 a(n) = -105*a(n-6) + 14*Sum_{k=1..5} a(n-k). - _Wesley Ivan Hurt_, May 11 2021
%p A163964 seq(coeff(series((1+t)*(1-t^6)/(1-15*t+119*t^6-105*t^7), t, n+1), t, n), n = 0 .. 30); # _G. C. Greubel_, Aug 11 2019
%t A163964 CoefficientList[Series[(1+t)*(1-t^6)/(1-15*t+119*t^6-105*t^7), {t,0,30}], t] (* _G. C. Greubel, Aug 23 2017 *)
%t A163964 coxG[{6, 105, -14}] (* The coxG program is at A169452 *) (* _G. C. Greubel_, Aug 11 2019 *)
%o A163964 (PARI) my(t='t+O('t^30)); Vec((1+t)*(1-t^6)/(1-15*t+119*t^6-105*t^7)) \\ _G. C. Greubel_, Aug 23 2017
%o A163964 (Magma) R<t>:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1+t)*(1-t^6)/(1-15*t+119*t^6-105*t^7) )); // _G. C. Greubel_, Aug 11 2019
%o A163964 (Sage)
%o A163964 def A163964_list(prec):
%o A163964     P.<t> = PowerSeriesRing(ZZ, prec)
%o A163964     return P((1+t)*(1-t^6)/(1-15*t+119*t^6-105*t^7)).list()
%o A163964 A163964_list(30) # _G. C. Greubel_, Aug 10 2019
%o A163964 (GAP) a:=[16, 240, 3600, 54000, 810000, 12149880];; for n in [7..30] do a[n]:=14*(a[n-1] +a[n-2]+a[n-3]+a[n-4]+a[n-5]) -105*a[n-6]; od; Concatenation([1], a); # _G. C. Greubel_, Aug 11 2019
%K A163964 nonn
%O A163964 0,2
%A A163964 _John Cannon_ and _N. J. A. Sloane_, Dec 03 2009