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A165912 Number of alternating polynomials of degree 3n in GF(2)[X], n>0.

%I A165912 #27 Dec 16 2017 05:23:55
%S A165912 2,0,2,2,4,6,12,20,38,66,124,224,420,774,1456,2720,5140,9690,18396,
%T A165912 34918,66576,127038,243148,465920,894784,1720530,3314018,6390930,
%U A165912 12341860,23860200,46182444,89477120,173534032,336857610,654471204,1272578048,2476377540,4822410222,9397535280
%N A165912 Number of alternating polynomials of degree 3n in GF(2)[X], n>0.
%C A165912 We define an alternating polynomial as follows: let I be the set of irreducible polynomials of degree > 1 over GF(2) and Sym_3 the symmetric group on 3 elements. For a polynomial P in I of degree n, we define P*(X) = X^n P(1/X) and P+(X) = P(X+1). The operators define an action of the group Sym_3 over I. Then an alternating polynomial is defined by the property that P*=P+.
%C A165912 The degree of an alternating polynomial is always 0 mod 3. The numbers in the sequence are always even. These polynomials are invariant under the action of the alternating subgroup Alt_3 of S3.
%H A165912 G. C. Greubel, <a href="/A165912/b165912.txt">Table of n, a(n) for n = 1..3300</a>
%H A165912 J.-F. Michon, P. Ravache, <a href="http://dx.doi.org/10.1016/j.ffa.2010.01.004"> On different families of invariant irreducible polynomials over F_2</a>, Finite fields & Applications 16 (2010) 163-174
%F A165912 a(n) = 2*(sum_{d|n, n/d != 0 mod 3} mu(n/d)*(2^d - (-1)^d))/(3n).
%F A165912 a(n) = 2 * A165920(n).
%t A165912 a[n_] := 2*DivisorSum[n, Boole[Mod[n/#, 3] != 0] MoebiusMu[n/#]*(2^# - (-1)^#) &]/(3 n); Array[a, 40] (* _Jean-François Alcover_, Dec 03 2015, adapted from PARI *)
%o A165912 (PARI) L(n, k) = sumdiv(gcd(n,k), d, moebius(d) * binomial(n/d, k/d) );
%o A165912 a(n) = sum(k=0, n, if( (n+k)%3!=0, L(n, k), 0 ) ) / n;
%o A165912 vector(55,n,a(n))
%o A165912 /* _Joerg Arndt_, Jun 28 2012 */
%Y A165912 A001037 is the enumeration by degree of the polynomials of the set I.
%Y A165912 A000048 is the enumeration by degree of the polynomials such that P=P* (self-reciprocal polynomials) which is the same as the one for the polynomials such that P=P+ or P=((P+)*)+.
%K A165912 easy,nonn
%O A165912 1,1
%A A165912 _Jean Francis Michon_ and Philippe Ravache (philippe.ravache(AT)univ-rouen.fr), Sep 30 2009
%E A165912 Edited by _N. J. A. Sloane_, May 15 2010