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A163219 Number of reduced words of length n in Coxeter group on 36 generators S_i with relations (S_i)^2 = (S_i S_j)^4 = I.

%I A163219 #21 Sep 08 2022 08:45:46
%S A163219 1,36,1260,44100,1542870,53978400,1888472880,66069561600,
%T A163219 2311490430270,80869130653500,2829263840578980,98983800307381500,
%U A163219 3463018394666864670,121156152466965222600,4238733846520797445080,148295107229819712107400
%N A163219 Number of reduced words of length n in Coxeter group on 36 generators S_i with relations (S_i)^2 = (S_i S_j)^4 = I.
%C A163219 The initial terms coincide with those of A170755, although the two sequences are eventually different.
%C A163219 Computed with MAGMA using commands similar to those used to compute A154638.
%H A163219 G. C. Greubel, <a href="/A163219/b163219.txt">Table of n, a(n) for n = 0..645</a>
%H A163219 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (34, 34, 34, -595).
%F A163219 G.f.: (t^4 + 2*t^3 + 2*t^2 + 2*t + 1)/(595*t^4 - 34*t^3 - 34*t^2 - 34*t + 1).
%F A163219 a(n) = -595*a(n-4) + 34*Sum_{k=1..3} a(n-k). - _Wesley Ivan Hurt_, May 05 2021
%t A163219 CoefficientList[Series[(t^4+2*t^3+2*t^2+2*t+1)/(595*t^4-34*t^3-34*t^2 - 34*t+1), {t, 0, 20}], t] (* or *) Join[{1}, LinearRecurrence[{34, 34, 34, -595}, {36, 1260, 44100, 1542870}, 20]] (* _G. C. Greubel_, Dec 11 2016 *)
%t A163219 coxG[{4, 595, -34}] (* The coxG program is at A169452 *) (* _G. C. Greubel_, Apr 30 2019 *)
%o A163219 (PARI) my(t='t+O('t^20)); Vec((t^4+2*t^3+2*t^2+2*t+1)/(595*t^4-34*t^3 - 34*t^2-34*t+1)) \\ _G. C. Greubel_, Dec 11 2016
%o A163219 (Magma) R<x>:=PowerSeriesRing(Integers(), 20); Coefficients(R!( (1+x)*(1-x^4)/(1-35*x+629*x^4-595*x^5) )); // _G. C. Greubel_, Apr 30 2019
%o A163219 (Sage) ((1+x)*(1-x^4)/(1-35*x+629*x^4-595*x^5)).series(x, 20).coefficients(x, sparse=False) # _G. C. Greubel_, Apr 30 2019
%Y A163219 Cf. A154638, A170755.
%K A163219 nonn,easy
%O A163219 0,2
%A A163219 _John Cannon_ and _N. J. A. Sloane_, Dec 03 2009