|
%I #37 Apr 30 2019 12:07:07
%S 1,0,0,1,1,2,5,6,15,24,45,85,155,288,550,1008,1935,3626,6885,13107,
%T 24940,47616,91225,174590,335626,645120,1242600,2396745,4627915,
%U 8947294,17318945,33552384,65076240,126320640,245424829,477218560,928638035,1808400384,3524082400,6871921458,13408691175,26178823218
%N Number of degree-n irreducible polynomials over GF(2) with trace = 0 and subtrace = 0.
%H Vincenzo Librandi, <a href="/A042980/b042980.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_{ L(n, k) : n+k = 2 mod 4}, where L(n, k) = Sum_{ mu(d)*binomial(n/d, k/d): d|gcd(n, k)}.
%t L[n_, k_] := Sum[ MoebiusMu[d]*Binomial[n/d, k/d], {d, Divisors[GCD[n,k]]}]/n;
%t a[n_]:=Sum[ If[ Mod[n+k, 4]==2, L[n, k], 0], {k, 0, n}];
%t 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==2, L(n, k), 0 ) ) / n;
%o vector(33,n,a(n))
%o /* _Joerg Arndt_, Jun 28 2012 */
%Y Cf. A042979, A042981, A042982.
%Y Cf. A074027, A074028, A074029, A074030.
%K nonn,nice,easy
%O 1,6
%A _Frank Ruskey_
|
