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A163223 Number of reduced words of length n in Coxeter group on 40 generators S_i with relations (S_i)^2 = (S_i S_j)^4 = I.

%I A163223 #20 Jun 10 2025 07:57:57
%S A163223 1,40,1560,60840,2371980,92476800,3605409600,140564736000,
%T A163223 5480222014020,213658376756760,8329936604744040,324760699264187160,
%U A163223 12661502336823753660,493636212105145265520,19245481572342746507280
%N A163223 Number of reduced words of length n in Coxeter group on 40 generators S_i with relations (S_i)^2 = (S_i S_j)^4 = I.
%C A163223 The initial terms coincide with those of A170759, although the two sequences are eventually different.
%C A163223 Computed with MAGMA using commands similar to those used to compute A154638.
%H A163223 G. C. Greubel, <a href="/A163223/b163223.txt">Table of n, a(n) for n = 0..625</a>
%H A163223 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (38, 38, 38, -741).
%F A163223 G.f.: (t^4 + 2*t^3 + 2*t^2 + 2*t + 1)/(741*t^4 - 38*t^3 - 38*t^2 - 38*t + 1).
%F A163223 a(n) = 38*a(n-1)+38*a(n-2)+38*a(n-3)-741*a(n-4). - _Wesley Ivan Hurt_, May 06 2021
%t A163223 CoefficientList[Series[(t^4+2*t^3+2*t^2+2*t+1)/(741*t^4-38*t^3-38*t^2 - 38*t+1), {t, 0, 20}], t] (* or *) LinearRecurrence[{38, 38, 38, -741}, {1, 40, 1560, 60840, 2371980}, 20] (* _G. C. Greubel_, Dec 11 2016 *)
%t A163223 coxG[{4, 741, -38}] (* The coxG program is at A169452 *) (* _G. C. Greubel_, Apr 30 2019 *)
%o A163223 (PARI) my(t='t+O('t^20)); Vec((t^4+2*t^3+2*t^2+2*t+1)/(741*t^4- 38*t^3 -38*t^2-38*t+1)) \\ _G. C. Greubel_, Dec 11 2016
%o A163223 (Magma) R<x>:=PowerSeriesRing(Integers(), 20); Coefficients(R!( (1+x)*(1-x^4)/(1-39*x+779*x^4-741*x^5) )); // _G. C. Greubel_, Apr 30 2019
%o A163223 (Sage) ((1+x)*(1-x^4)/(1-39*x+779*x^4-741*x^5)).series(x, 20).coefficients(x, sparse=False) # _G. C. Greubel_, Apr 30 2019
%Y A163223 Cf. A154638, A170759.
%K A163223 nonn
%O A163223 0,2
%A A163223 _John Cannon_ and _N. J. A. Sloane_, Dec 03 2009