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A164070 Number of reduced words of length n in Coxeter group on 37 generators S_i with relations (S_i)^2 = (S_i S_j)^6 = I.

%I A164070 #17 Sep 08 2022 08:45:47
%S A164070 1,37,1332,47952,1726272,62145792,2237247846,80540898480,
%T A164070 2899471482810,104380942332240,3757712806199520,135277620783782400,
%U A164070 4869992899598197770,175319692235303773500,6311507043063165819750
%N A164070 Number of reduced words of length n in Coxeter group on 37 generators S_i with relations (S_i)^2 = (S_i S_j)^6 = I.
%C A164070 The initial terms coincide with those of A170756, although the two sequences are eventually different.
%C A164070 Computed with MAGMA using commands similar to those used to compute A154638.
%H A164070 G. C. Greubel, <a href="/A164070/b164070.txt">Table of n, a(n) for n = 0..640</a>
%H A164070 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (35,35,35,35,35,-630).
%F A164070 G.f.: (t^6 + 2*t^5 + 2*t^4 + 2*t^3 + 2*t^2 + 2*t + 1)/(630*t^6 - 35*t^5 - 35*t^4 - 35*t^3 - 35*t^2 - 35*t + 1).
%F A164070 a(n) = -630*a(n-6) + 35*Sum_{k=1..5} a(n-k). - _Wesley Ivan Hurt_, May 05 2021
%p A164070 seq(coeff(series((1+t)*(1-t^6)/(1-36*t+665*t^6-630*t^7), t, n+1), t, n), n = 0 .. 30); # _G. C. Greubel_, Aug 16 2019
%t A164070 CoefficientList[Series[(1+t)*(1-t^6)/(1-36*t+665*t^6-630*t^7), {t,0,30}], t] (* _G. C. Greubel_, Sep 09 2017 *)
%t A164070 coxG[{6, 630, -35}] (* The coxG program is at A169452 *) (* _G. C. Greubel_, Aug 16 2019 *)
%o A164070 (PARI) my(t='t+O('t^30)); Vec((1+t)*(1-t^6)/(1-36*t+665*t^6-630*t^7)) \\ _G. C. Greubel_, Sep 09 2017
%o A164070 (Magma) R<t>:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1+t)*(1-t^6)/(1-36*t+665*t^6-630*t^7) )); // _G. C. Greubel_, Aug 16 2019
%o A164070 (Sage)
%o A164070 def A164070_list(prec):
%o A164070     P.<t> = PowerSeriesRing(ZZ, prec)
%o A164070     return P((1+t)*(1-t^6)/(1-36*t+665*t^6-630*t^7)).list()
%o A164070 A164070_list(30) # _G. C. Greubel_, Aug 16 2019
%o A164070 (GAP) a:=[37, 1332, 47952, 1726272, 62145792, 2237247846];; for n in [7..30] do a[n]:=35*(a[n-1] +a[n-2]+a[n-3]+a[n-4]+a[n-5]) -630*a[n-6]; od; Concatenation([1], a); # _G. C. Greubel_, Aug 16 2019
%K A164070 nonn
%O A164070 0,2
%A A164070 _John Cannon_ and _N. J. A. Sloane_, Dec 03 2009