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A164026 Number of reduced words of length n in Coxeter group on 29 generators S_i with relations (S_i)^2 = (S_i S_j)^6 = I.

%I A164026 #19 Sep 08 2022 08:45:47
%S A164026 1,29,812,22736,636608,17825024,499100266,13974796080,391293972342,
%T A164026 10956222324432,306773975852064,8589664345360896,240510406272356430,
%U A164026 6734285904493468188,188559852134231228994,5279671570397554562148
%N A164026 Number of reduced words of length n in Coxeter group on 29 generators S_i with relations (S_i)^2 = (S_i S_j)^6 = I.
%C A164026 The initial terms coincide with those of A170748, although the two sequences are eventually different.
%C A164026 Computed with MAGMA using commands similar to those used to compute A154638.
%H A164026 G. C. Greubel, <a href="/A164026/b164026.txt">Table of n, a(n) for n = 0..685</a>
%H A164026 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (27,27,27,27,27,-378).
%F A164026 G.f.: (t^6 + 2*t^5 + 2*t^4 + 2*t^3 + 2*t^2 + 2*t + 1)/(378*t^6 - 27*t^5 - 27*t^4 - 27*t^3 - 27*t^2 - 27*t + 1).
%F A164026 a(n) = -378*a(n-6) + 27*Sum_{k=1..5} a(n-k). - _Wesley Ivan Hurt_, May 11 2021
%p A164026 seq(coeff(series((1+t)*(1-t^6)/(1-28*t+405*t^6-378*t^7), t, n+1), t, n), n = 0 .. 30); # _G. C. Greubel_, Aug 13 2019
%t A164026 CoefficientList[Series[(1+t)*(1-t^6)/(1-28*t+405*t^6-378*t^7), {t,0,30}], t] (* _G. C. Greubel_, Sep 07 2017 *)
%t A164026 coxG[{6,378,-27}] (* The coxG program is at A169452 *) (* _Harvey P. Dale_, Apr 14 2019 *)
%o A164026 (PARI) my(t='t+O('t^30)); Vec((1+t)*(1-t^6)/(1-28*t+405*t^6-378*t^7)) \\ _G. C. Greubel_, Sep 07 2017
%o A164026 (Magma) R<t>:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1+t)*(1-t^6)/(1-28*t+405*t^6-378*t^7) )); // _G. C. Greubel_, Aug 13 2019
%o A164026 (Sage)
%o A164026 def A164026_list(prec):
%o A164026     P.<t> = PowerSeriesRing(ZZ, prec)
%o A164026     return P((1+t)*(1-t^6)/(1-28*t+405*t^6-378*t^7)).list()
%o A164026 A164026_list(30) # _G. C. Greubel_, Aug 13 2019
%o A164026 (GAP) a:=[29, 812, 22736, 636608, 17825024, 499100266];; for n in [7..30] do a[n]:=27*(a[n-1] +a[n-2]+a[n-3]+a[n-4]+a[n-5]) -378*a[n-6]; od; Concatenation([1], a); # _G. C. Greubel_, Aug 13 2019
%K A164026 nonn
%O A164026 0,2
%A A164026 _John Cannon_ and _N. J. A. Sloane_, Dec 03 2009