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A027377 Number of irreducible polynomials of degree n over GF(4); dimensions of free Lie algebras.

%I A027377 #50 Dec 16 2024 04:31:00
%S A027377 1,4,6,20,60,204,670,2340,8160,29120,104754,381300,1397740,5162220,
%T A027377 19172790,71582716,268431360,1010580540,3817733920,14467258260,
%U A027377 54975528948,209430785460,799644629550,3059510616420
%N A027377 Number of irreducible polynomials of degree n over GF(4); dimensions of free Lie algebras.
%C A027377 Apart from initial terms, exponents in expansion of A065419 as a product zeta(n)^(-a(n)).
%C A027377 Number of aperiodic necklaces with n beads of 4 colors. - _Herbert Kociemba_, Nov 25 2016
%D A027377 E. R. Berlekamp, Algebraic Coding Theory, McGraw-Hill, NY, 1968, p. 84.
%D A027377 M. Lothaire, Combinatorics on Words. Addison-Wesley, Reading, MA, 1983, p. 79.
%H A027377 Seiichi Manyama, <a href="/A027377/b027377.txt">Table of n, a(n) for n = 0..1666</a> (terms 0..200 from T. D. Noe)
%H A027377 E. N. Gilbert and J. Riordan, <a href="http://projecteuclid.org/euclid.ijm/1255631587">Symmetry types of periodic sequences</a>, Illinois J. Math., 5 (1961), 657-665.
%H A027377 G. Niklasch, <a href="/A001692/a001692.html">Some number theoretical constants: 1000-digit values</a> [Cached copy]
%H A027377 A. Pakapongpun and T. Ward, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL12/Ward/ward17.html">Functorial Orbit counting</a>, JIS 12 (2009) 09.2.4, example 3.
%H A027377 Y. Puri and T. Ward, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL4/WARD/short.html">Arithmetic and growth of periodic orbits</a>, J. Integer Seqs., Vol. 4 (2001), #01.2.1.
%H A027377 G. Viennot, <a href="http://dx.doi.org/10.1007/BFb0067950">Algèbres de Lie Libres et Monoïdes Libres</a>, Lecture Notes in Mathematics 691, Springer Verlag 1978.
%H A027377 <a href="/index/Lu#Lyndon">Index entries for sequences related to Lyndon words</a>
%F A027377 a(n) = Sum_{d|n} mu(d)*4^(n/d)/n.
%F A027377 G.f.: k=4, 1 - Sum_{i>=1} mu(i)*log(1 - k*x^i)/i. - _Herbert Kociemba_, Nov 25 2016
%F A027377 a(n) = A054661(n) + 3 * A054660(n). - _Andrey Zabolotskiy_, Dec 17 2020
%F A027377 a(n) = 2 * (A054664(n) + A054660(n)). - _Andrey Zabolotskiy_, Dec 19 2020
%F A027377 a(n) = A054719(n)/n, n>0. - _R. J. Mathar_, Dec 16 2024
%p A027377 A027377 := proc(n) local d,s; if n = 0 then RETURN(1); else s := 0; for d in divisors(n) do s := s+mobius(d)*4^(n/d); od; RETURN(s/n); fi; end;
%t A027377 a[n_] := Sum[MoebiusMu[d]*4^(n/d), {d, Divisors[n]}] / n; a[0] = 1; Table[a[n], {n, 0, 23}](* _Jean-François Alcover_, Nov 29 2011 *)
%t A027377 mx=40;f[x_,k_]:=1-Sum[MoebiusMu[i] Log[1-k*x^i]/i,{i,1,mx}];CoefficientList[Series[f[x,4],{x,0,mx}],x] (* _Herbert Kociemba_, Nov 25 2016 *)
%o A027377 (PARI) a(n)=if(n,sumdiv(n,d,moebius(d)<<(2*n/d))/n,1) \\ _Charles R Greathouse IV_, Nov 29 2011
%Y A027377 Cf. A001037, A027376, A054719, A054660, A054661.
%Y A027377 Column k=4 of A074650.
%K A027377 nonn,nice,easy
%O A027377 0,2
%A A027377 _N. J. A. Sloane_