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A163957 Number of reduced words of length n in Coxeter group on 12 generators S_i with relations (S_i)^2 = (S_i S_j)^6 = I.

%I A163957 #20 Sep 08 2022 08:45:47
%S A163957 1,12,132,1452,15972,175692,1932546,21257280,233822160,2571956640,
%T A163957 28290564720,311185670400,3422926421970,37650915208500,
%U A163957 414146038003500,4555452101075700,50108275682741100,551172361422635700
%N A163957 Number of reduced words of length n in Coxeter group on 12 generators S_i with relations (S_i)^2 = (S_i S_j)^6 = I.
%C A163957 The initial terms coincide with those of A003954, although the two sequences are eventually different.
%C A163957 Computed with MAGMA using commands similar to those used to compute A154638.
%H A163957 G. C. Greubel, <a href="/A163957/b163957.txt">Table of n, a(n) for n = 0..955</a>
%H A163957 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (10,10,10,10,10,-55).
%F A163957 G.f.: (t^6 + 2*t^5 + 2*t^4 + 2*t^3 + 2*t^2 + 2*t + 1)/(55*t^6 - 10*t^5 - 10*t^4 - 10*t^3 - 10*t^2 - 10*t + 1).
%F A163957 a(n) = -55*a(n-6) + 10*Sum_{k=1..5} a(n-k). - _Wesley Ivan Hurt_, May 11 2021
%p A163957 seq(coeff(series((1+t)*(1-t^6)/(1-11*t+65*t^6-55*t^7), t, n+1), t, n), n = 0 .. 30); # _G. C. Greubel_, Aug 10 2019
%t A163957 CoefficientList[Series[(1+t)*(1-t^6)/(1-11*t+65*t^6-55*t^7), {t,0,30}], t] (* _G. C. Greubel_, Aug 13 2017 *)
%t A163957 coxG[{6, 55, -10}] (* The coxG program is at A169452 *) (* _G. C. Greubel_, Aug 10 2019 *)
%o A163957 (PARI) my(t='t+O('t^30)); Vec((1+t)*(1-t^6)/(1-11*t+65*t^6-55*t^7)) \\ _G. C. Greubel_, Aug 13 2017
%o A163957 (Magma) R<t>:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1+t)*(1-t^6)/(1-11*t+65*t^6-55*t^7) )); // _G. C. Greubel_, Aug 10 2019
%o A163957 (Sage)
%o A163957 def A163957_list(prec):
%o A163957     P.<t> = PowerSeriesRing(ZZ, prec)
%o A163957     return P((1+t)*(1-t^6)/(1-11*t+65*t^6-55*t^7)).list()
%o A163957 A163957_list(30) # _G. C. Greubel_, Aug 10 2019
%o A163957 (GAP) a:=[12,132,1452,15972,175692,1932546];; for n in [7..30] do a[n]:=10*(a[n-1] +a[n-2]+a[n-3]+a[n-4]+a[n-5]) -55*a[n-6]; od; Concatenation([1], a); # _G. C. Greubel_, Aug 10 2019
%K A163957 nonn
%O A163957 0,2
%A A163957 _John Cannon_ and _N. J. A. Sloane_, Dec 03 2009