

A165912


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


2



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
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OFFSET

1,1


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



FORMULA

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


MATHEMATICA

a[n_] := 2*DivisorSum[n, Boole[Mod[n/#, 3] != 0] MoebiusMu[n/#]*(2^#  (1)^#) &]/(3 n); Array[a, 40] (* JeanFrançois Alcover, Dec 03 2015, adapted from PARI *)


PROG

(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))


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* (selfreciprocal polynomials) which is the same as the one for the polynomials such that P=P+ or P=((P+)*)+.


KEYWORD

easy,nonn


AUTHOR



EXTENSIONS



STATUS

approved



