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A163977 Number of reduced words of length n in Coxeter group on 21 generators S_i with relations (S_i)^2 = (S_i S_j)^6 = I.

%I A163977 #15 Sep 08 2022 08:45:47
%S A163977 1,21,420,8400,168000,3360000,67199790,1343991600,26879748210,
%T A163977 537593288400,10751832252000,215035974720000,4300706088043890,
%U A163977 86013853634593500,1720271710182898110,34405326953812846500
%N A163977 Number of reduced words of length n in Coxeter group on 21 generators S_i with relations (S_i)^2 = (S_i S_j)^6 = I.
%C A163977 The initial terms coincide with those of A170740, although the two sequences are eventually different.
%C A163977 Computed with MAGMA using commands similar to those used to compute A154638.
%H A163977 G. C. Greubel, <a href="/A163977/b163977.txt">Table of n, a(n) for n = 0..765</a>
%H A163977 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (19,19,19,19,19,-190).
%F A163977 G.f.: (t^6 + 2*t^5 + 2*t^4 + 2*t^3 + 2*t^2 + 2*t + 1)/(190*t^6 - 19*t^5 - 19*t^4 - 19*t^3 - 19*t^2 - 19*t + 1).
%F A163977 a(n) = -190*a(n-6) + 19*Sum_{k=1..5} a(n-k). - _Wesley Ivan Hurt_, May 11 2021
%p A163977 seq(coeff(series((1+t)*(1-t^6)/(1-20*t+209*t^6-190*t^7), t, n+1), t, n), n = 0 .. 30); # _G. C. Greubel_, Aug 11 2019
%t A163977 CoefficientList[Series[(1+t)*(1-t^6)/(1-20*t+209*t^6-190*t^7), {t,0,30}], t] (* _G. C. Greubel_, Aug 24 2017 *)
%t A163977 coxG[{6, 190, -19}] (* The coxG program is at A169452 *) (* _G. C. Greubel_, Aug 11 2019 *)
%o A163977 (PARI) my(t='t+O('t^30)); Vec((1+t)*(1-t^6)/(1-20*t+209*t^6-190*t^7)) \\ _G. C. Greubel_, Aug 24 2017
%o A163977 (Magma) R<t>:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1+t)*(1-t^6)/(1-20*t+209*t^6-190*t^7) )); // _G. C. Greubel_, Aug 11 2019
%o A163977 (Sage)
%o A163977 def A163977_list(prec):
%o A163977     P.<t> = PowerSeriesRing(ZZ, prec)
%o A163977     return P((1+t)*(1-t^6)/(1-20*t+209*t^6-190*t^7)).list()
%o A163977 A163977_list(30) # _G. C. Greubel_, Aug 11 2019
%o A163977 (GAP) a:=[21, 420, 8400, 168000, 3360000, 67199790];; for n in [7..30] do a[n]:=19*(a[n-1] +a[n-2]+a[n-3]+a[n-4]+a[n-5]) -190*a[n-6]; od; Concatenation([1], a); # _G. C. Greubel_, Aug 11 2019
%K A163977 nonn
%O A163977 0,2
%A A163977 _John Cannon_ and _N. J. A. Sloane_, Dec 03 2009