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

2, 0, 2, 2, 4, 6, 12, 20, 38, 66, 124, 224, 420, 774, 1456, 2720, 5140, 9690, 18396, 34918, 66576, 127038, 243148, 465920, 894784, 1720530, 3314018, 6390930, 12341860, 23860200, 46182444, 89477120, 173534032, 336857610, 654471204, 1272578048, 2476377540, 4822410222, 9397535280

Offset: 1

Author

Jean Francis Michon and Philippe Ravache (philippe.ravache(AT)univ-rouen.fr), Sep 30 2009

Keywords

• easy
• nonn

Comments

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+.

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.

Links

• G. C. Greubel, Table of n, a(n) for n = 1..3300
• J.-F. Michon, P. Ravache, On different families of invariant irreducible polynomials over F_2, Finite fields & Applications 16 (2010) 163-174

Crossrefs

A001037 is the enumeration by degree of the polynomials of the set I.

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+)*)+.

Programs

Mathematica

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 *)

PARI

L(n, k) = sumdiv(gcd(n,k), d, moebius(d) * binomial(n/d, k/d) );
a(n) = sum(k=0, n, if( (n+k)%3!=0, L(n, k), 0 ) ) / n;
vector(55,n,a(n))
/* Joerg Arndt, Jun 28 2012 */

Formula

a(n) = 2*(sum_{d|n, n/d != 0 mod 3} mu(n/d)*(2^d - (-1)^d))/(3n).

a(n) = 2 * A165920(n).

Extensions

Edited by N. J. A. Sloane, May 15 2010