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A165777 Number of reduced words of length n in Coxeter group on 6 generators S_i with relations (S_i)^2 = (S_i S_j)^10 = I.

%I A165777 #22 Sep 08 2022 08:45:48
%S A165777 1,6,30,150,750,3750,18750,93750,468750,2343750,11718735,58593600,
%T A165777 292967640,1464836400,7324173000,36620820000,183103875000,
%U A165777 915518250000,4577585625000,22887900000000,114439359375210,572196093753000,2860976953154040,14304867187737000
%N A165777 Number of reduced words of length n in Coxeter group on 6 generators S_i with relations (S_i)^2 = (S_i S_j)^10 = I.
%C A165777 The initial terms coincide with those of A003948, although the two sequences are eventually different.
%C A165777 Computed with MAGMA using commands similar to those used to compute A154638.
%H A165777 G. C. Greubel, <a href="/A165777/b165777.txt">Table of n, a(n) for n = 0..500</a>
%H A165777 <a href="/index/Rec#order_10">Index entries for linear recurrences with constant coefficients</a>, signature (4,4,4,4,4,4,4,4,4,-10).
%F A165777 G.f.: (t^10 + 2*t^9 + 2*t^8 + 2*t^7 + 2*t^6 + 2*t^5 + 2*t^4 + 2*t^3 + 2*t^2 + 2*t + 1)/(10*t^10 - 4*t^9 - 4*t^8 - 4*t^7 - 4*t^6 - 4*t^5 - 4*t^4 - 4*t^3 - 4*t^2 - 4*t + 1).
%p A165777 A165777 := proc(n)
%p A165777 coeftayl( (t^10 + 2*t^9 + 2*t^8 + 2*t^7 + 2*t^6 + 2*t^5 + 2*t^4 + 2*t^3 + 2*t^2 + 2*t + 1)/(10*t^10 - 4*t^9 - 4*t^8 - 4*t^7 - 4*t^6 - 4*t^5 - 4*t^4 - 4*t^3 - 4*t^2 - 4*t + 1), t=0, n);
%p A165777 end proc:
%p A165777 seq(A165777(n), n=0..25); # _Wesley Ivan Hurt_, Nov 14 2014
%t A165777 CoefficientList[Series[(t^10 + 2*t^9 + 2*t^8 + 2*t^7 + 2*t^6 + 2*t^5 + 2*t^4 + 2*t^3 + 2*t^2 + 2*t + 1)/(10*t^10 - 4*t^9 - 4*t^8 - 4*t^7 - 4*t^6 - 4*t^5 - 4*t^4 - 4*t^3 - 4*t^2 - 4*t + 1), {t, 0, 25}], t] (* _Wesley Ivan Hurt_, Nov 14 2014 *)
%t A165777 coxG[{10, 10, -4}] (* The coxG program is at A169452 *) (* _G. C. Greubel_, Sep 17 2019 *)
%o A165777 (PARI) my(t='t+O('t^30)); Vec((1+t)*(1-t^10)/(1-5*t+14*t^10-10*t^11)) \\ _G. C. Greubel_, Sep 17 2019
%o A165777 (Magma) R<t>:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1+t)*(1-t^10)/(1-5*t+14*t^10-10*t^11) )); // _G. C. Greubel_, Sep 17 2019
%o A165777 (Sage)
%o A165777 def A165777_list(prec):
%o A165777     P.<t> = PowerSeriesRing(ZZ, prec)
%o A165777     return P((1+t)*(1-t^10)/(1-5*t+14*t^10-10*t^11)).list()
%o A165777 A165777_list(30) # _G. C. Greubel_, Sep 17 2019
%o A165777 (GAP) a:=[6, 30, 150, 750, 3750, 18750, 93750, 468750, 2343750, 11718735];; for n in [11..30] do a[n]:=4*Sum([1..9], j-> a[n-j]) -10*a[n-10]; od; Concatenation([1], a); # _G. C. Greubel_, Sep 17 2019
%Y A165777 Cf. A003948, A154638.
%K A165777 nonn
%O A165777 0,2
%A A165777 _John Cannon_ and _N. J. A. Sloane_, Dec 03 2009
%E A165777 More terms from _Wesley Ivan Hurt_, Nov 14 2014