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A163995 Number of reduced words of length n in Coxeter group on 26 generators S_i with relations (S_i)^2 = (S_i S_j)^6 = I.

%I A163995 #16 Sep 08 2022 08:45:47
%S A163995 1,26,650,16250,406250,10156250,253905925,6347640000,158690797200,
%T A163995 3967264860000,99181494750000,2479534200000000,61988275781355300,
%U A163995 1549704914070300000,38742573340231207200,968563095719204700000,24214046448355276500000,605350387594249537500000
%N A163995 Number of reduced words of length n in Coxeter group on 26 generators S_i with relations (S_i)^2 = (S_i S_j)^6 = I.
%C A163995 The initial terms coincide with those of A170745, although the two sequences are eventually different.
%C A163995 Computed with MAGMA using commands similar to those used to compute A154638.
%H A163995 G. C. Greubel, <a href="/A163995/b163995.txt">Table of n, a(n) for n = 0..710</a>
%H A163995 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (24,24,24,24,24,-300).
%F A163995 G.f.: (t^6 + 2*t^5 + 2*t^4 + 2*t^3 + 2*t^2 + 2*t + 1)/(300*t^6 - 24*t^5 - 24*t^4 - 24*t^3 - 24*t^2 - 24*t + 1).
%F A163995 a(n) = -300*a(n-6) + 24*Sum_{k=1..5} a(n-k). - _Wesley Ivan Hurt_, May 11 2021
%p A163995 seq(coeff(series((1+t)*(1-t^6)/(1-25*t+324*t^6-300*t^7), t, n+1), t, n), n = 0 .. 30); # _G. C. Greubel_, Aug 13 2019
%t A163995 CoefficientList[Series[(1+t)*(1-t^6)/(1-25*t+324*t^6-300*t^7), {t,0,30}], t] (* _G. C. Greubel_, Aug 24 2017 *)
%t A163995 coxG[{6, 300, -24}] (* The coxG program is at A169452 *) (* _G. C. Greubel_, Aug 13 2019 *)
%o A163995 (PARI) my(t='t+O('t^30)); Vec((1+t)*(1-t^6)/(1-25*t+324*t^6-300*t^7)) \\ _G. C. Greubel_, Aug 24 2017
%o A163995 (Magma) R<t>:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1+t)*(1-t^6)/(1-25*t+324*t^6-300*t^7) )); // _G. C. Greubel_, Aug 13 2019
%o A163995 (Sage)
%o A163995 def A163995_list(prec):
%o A163995     P.<t> = PowerSeriesRing(ZZ, prec)
%o A163995     return P((1+t)*(1-t^6)/(1-25*t+324*t^6-300*t^7)).list()
%o A163995 A163995_list(30) # _G. C. Greubel_, Aug 13 2019
%o A163995 (GAP) a:=[26, 650, 16250, 406250, 10156250, 253905925];; for n in [7..30] do a[n]:=24*(a[n-1] +a[n-2]+a[n-3]+a[n-4]+a[n-5]) -300*a[n-6]; od; Concatenation([1], a); # _G. C. Greubel_, Aug 13 2019
%K A163995 nonn
%O A163995 0,2
%A A163995 _John Cannon_ and _N. J. A. Sloane_, Dec 03 2009