%I #38 Apr 30 2019 12:05:44

%S 0,0,1,0,2,2,4,8,13,24,48,80,160,288,541,1024,1920,3626,6912,13056,

%T 24989,47616,91136,174760,335462,645120,1242904,2396160,4628480,

%U 8947294,17317888,33554432,65074253,126320640,245428574,477211280,928645120,1808400384,3524068955,6871947672,13408665600,26178823218

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

%H Vincenzo Librandi, <a href="/A042979/b042979.txt">Table of n, a(n) for n = 1..1000</a>

%H K. Cattell, C. R. Miers, F. Ruskey, J. Sawada and M. Serra, <a href="https://www.researchgate.net/publication/2634456_The_Number_of_Irreducible_Polynomials_over_GF2_with_Given_Trace_and_Subtrace">The Number of Irreducible Polynomials over GF(2) with Given Trace and Subtrace</a>, J. Comb. Math. and Comb. Comp., 47 (2003) 31-64.

%H F. Ruskey, <a href="http://combos.org/TSpoly">Number of irreducible polynomials over GF(2) with given trace and subtrace</a>

%F a(n) = (1/n) * Sum_{k=0..n, n+k == 0 (mod 4)} L(n, k), where L(n, k) = Sum_{d|gcd(n, k)} mu(d)*binomial(n/d, k/d).

%t 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] == 0, L[n, k], 0], {k, 0, n}]; Table[a[n], {n, 1, 32}] (* _Jean-François Alcover_, Jun 28 2012, from formula *)

%o (PARI)

%o L(n, k) = sumdiv(gcd(n,k), d, moebius(d) * binomial(n/d, k/d) );

%o a(n) = sum(k=0, n, if( (n+k)%4==0, L(n, k), 0 ) ) / n;

%o vector(33,n,a(n))

%o /* _Joerg Arndt_, Jun 28 2012 */

%Y Cf. A042980, A042981, A042982.

%Y Cf. A074027, A074028, A074029, A074030.

%K nonn

%O 1,5

%A _Frank Ruskey_