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A164027 Number of reduced words of length n in Coxeter group on 30 generators S_i with relations (S_i)^2 = (S_i S_j)^6 = I.

%I A164027 #20 Sep 08 2022 08:45:47
%S A164027 1,30,870,25230,731670,21218430,615334035,17844674400,517495192200,
%T A164027 15007349977200,435212842037400,12621163507344000,366013483272687390,
%U A164027 10614383520144862920,307816904736225416280,8926683934263695000520
%N A164027 Number of reduced words of length n in Coxeter group on 30 generators S_i with relations (S_i)^2 = (S_i S_j)^6 = I.
%C A164027 The initial terms coincide with those of A170749, although the two sequences are eventually different.
%C A164027 Computed with MAGMA using commands similar to those used to compute A154638.
%H A164027 G. C. Greubel, <a href="/A164027/b164027.txt">Table of n, a(n) for n = 0..680</a>
%H A164027 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (28,28,28,28,28,-406).
%F A164027 G.f.: (t^6 + 2*t^5 + 2*t^4 + 2*t^3 + 2*t^2 + 2*t + 1)/(406*t^6 - 28*t^5 - 28*t^4 - 28*t^3 - 28*t^2 - 28*t + 1).
%F A164027 a(n) = -406*a(n-6) + 28*Sum_{k=1..5} a(n-k). - _Wesley Ivan Hurt_, May 11 2021
%p A164027 seq(coeff(series((1+t)*(1-t^6)/(1-29*t+434*t^6-406*t^7), t, n+1), t, n), n = 0 .. 30); # _G. C. Greubel_, Aug 10 2019
%t A164027 CoefficientList[Series[(1+t)*(1-t^6)/(1-29*t+434*t^6-406*t^7), {t,0,30}], t] (* _G. C. Greubel_, Sep 07 2017 *)
%t A164027 coxG[{6,406,-28}] (* The coxG program is at A169452 *) (* _Harvey P. Dale_, Jul 05 2019 *)
%o A164027 (PARI) my(t='t+O('t^30)); Vec((1+t)*(1-t^6)/(1-29*t+434*t^6-406*t^7)) \\ _G. C. Greubel_, Sep 07 2017
%o A164027 (Magma) R<t>:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1+t)*(1-t^6)/(1-29*t+434*t^6-406*t^7) )); // _G. C. Greubel_, Aug 10 2019
%o A164027 (Sage)
%o A164027 def A164027_list(prec):
%o A164027     P.<t> = PowerSeriesRing(ZZ, prec)
%o A164027     return P((1+t)*(1-t^6)/(1-29*t+434*t^6-406*t^7)).list()
%o A164027 A164027_list(30) # _G. C. Greubel_, Aug 10 2019
%o A164027 (GAP) a:=[30, 870, 25230, 731670, 21218430, 615334035];; for n in [7..30] do a[n]:=28*(a[n-1] +a[n-2]+a[n-3]+a[n-4]+a[n-5]) -406*a[n-6]; od; Concatenation([1], a); # _G. C. Greubel_, Aug 10 2019
%K A164027 nonn
%O A164027 0,2
%A A164027 _John Cannon_ and _N. J. A. Sloane_, Dec 03 2009