A042981 Number of degree-n irreducible polynomials over GF(2) with trace = 1 and subtrace = 0.

1, 0, 1, 1, 1, 3, 4, 8, 15, 24, 48, 85, 155, 297, 541, 1024, 1935, 3626, 6912, 13107, 24940, 47709, 91136, 174760, 335626, 645120, 1242904, 2396745, 4627915, 8948385, 17317888, 33554432, 65076240, 126320640, 245428574, 477218560, 928638035, 1808414181, 3524068955, 6871947672, 13408691175, 26178823218

Offset: 1

Author

Frank Ruskey

Keywords

• nonn

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..1000
• K. Cattell, C. R. Miers, F. Ruskey, J. Sawada and M. Serra, The Number of Irreducible Polynomials over GF(2) with Given Trace and Subtrace, J. Comb. Math. and Comb. Comp., 47 (2003) 31-64.
• F. Ruskey, Number of irreducible polynomials over GF(2) with given trace and subtrace

Crossrefs

Cf. A042979, A042980, A042982.

Cf. A074027, A074028, A074029, A074030.

Programs

Mathematica

L[n_, k_] := Sum[ MoebiusMu[d]*Binomial[n/d, k/d], {d, Divisors[GCD[n, k]]}]/n;
a[n_] := Sum[ If[ Mod[n+k, 4] == 1, L[n, k], 0], {k, 0, n}];
Table[a[n], {n, 1, 32}]
(* Jean-François Alcover, Jun 28 2012, from formula *)

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)%4==1, L(n, k), 0 ) ) / n;
vector(33,n,a(n))
/* Joerg Arndt, Jun 28 2012 */

Formula

a(n) = (1/n) * Sum_{ L(n, k) : n+k = 1 mod 4}, where L(n, k) = Sum_{ mu(d)*{binomial(n/d, k/d)} : d|gcd(n, k)}.