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A164017 Number of reduced words of length n in Coxeter group on 27 generators S_i with relations (S_i)^2 = (S_i S_j)^6 = I.

%I A164017 #19 Sep 08 2022 08:45:47
%S A164017 1,27,702,18252,474552,12338352,320796801,8340707700,216858163275,
%T A164017 5638306085100,146595798051300,3811486585140000,99098542944724050,
%U A164017 2576559301574090625,66990468651299212500,1741750282005552804375
%N A164017 Number of reduced words of length n in Coxeter group on 27 generators S_i with relations (S_i)^2 = (S_i S_j)^6 = I.
%C A164017 The initial terms coincide with those of A170746, although the two sequences are eventually different.
%C A164017 Computed with MAGMA using commands similar to those used to compute A154638.
%H A164017 G. C. Greubel, <a href="/A164017/b164017.txt">Table of n, a(n) for n = 0..705</a>
%H A164017 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (25,25,25,25,25,-325).
%F A164017 G.f.: (t^6 + 2*t^5 + 2*t^4 + 2*t^3 + 2*t^2 + 2*t + 1)/(325*t^6 - 25*t^5 - 25*t^4 - 25*t^3 - 25*t^2 - 25*t + 1).
%F A164017 a(n) = -325*a(n-6) + 25*Sum_{k=1..5} a(n-k). - _Wesley Ivan Hurt_, May 11 2021
%p A164017 seq(coeff(series((1+t)*(1-t^6)/(1-26*t+350*t^6-325*t^7), t, n+1), t, n), n = 0 .. 30); # _G. C. Greubel_, Aug 13 2019
%t A164017 coxG[{6,325,-25}] (* The coxG program is at A169452 *) (* _Harvey P. Dale_, Jun 26 2017 *)
%t A164017 CoefficientList[Series[(1+t)*(1-t^6)/(1-26*t+350*t^6-325*t^7), {t,0,30}], t] (* _G. C. Greubel_, Sep 07 2017 *)
%o A164017 (PARI) my(t='t+O('t^30)); Vec((1+t)*(1-t^6)/(1-26*t+350*t^6-325*t^7)) \\ _G. C. Greubel_, Sep 07 2017
%o A164017 (Magma) R<t>:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1+t)*(1-t^6)/(1-26*t+350*t^6-325*t^7) )); // _G. C. Greubel_, Aug 13 2019
%o A164017 (Sage)
%o A164017 def A164017_list(prec):
%o A164017     P.<t> = PowerSeriesRing(ZZ, prec)
%o A164017     return P((1+t)*(1-t^6)/(1-26*t+350*t^6-325*t^7)).list()
%o A164017 A164017_list(30) # _G. C. Greubel_, Aug 13 2019
%o A164017 (GAP) a:=[27, 702, 18252, 474552, 12338352, 320796801];; for n in [7..30] do a[n]:=25*(a[n-1] +a[n-2]+a[n-3]+a[n-4]+a[n-5]) -325*a[n-6]; od; Concatenation([1], a); # _G. C. Greubel_, Aug 13 2019
%K A164017 nonn
%O A164017 0,2
%A A164017 _John Cannon_ and _N. J. A. Sloane_, Dec 03 2009