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A042979
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Number of degree-n irreducible polynomials over GF(2) with trace = 0 and subtrace = 1.
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10
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0, 0, 1, 0, 2, 2, 4, 8, 13, 24, 48, 80, 160, 288, 541, 1024, 1920, 3626, 6912, 13056, 24989, 47616, 91136, 174760, 335462, 645120, 1242904, 2396160, 4628480, 8947294, 17317888, 33554432, 65074253, 126320640, 245428574, 477211280, 928645120, 1808400384, 3524068955, 6871947672, 13408665600, 26178823218
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OFFSET
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1,5
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LINKS
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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
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FORMULA
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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).
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MATHEMATICA
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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 *)
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PROG
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(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==0, L(n, k), 0 ) ) / n;
vector(33, n, a(n))
/* Joerg Arndt, Jun 28 2012 */
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CROSSREFS
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Cf. A042980, A042981, A042982.
Cf. A074027, A074028, A074029, A074030.
Sequence in context: A104700 A039941 A036761 * A000018 A306604 A075126
Adjacent sequences: A042976 A042977 A042978 * A042980 A042981 A042982
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KEYWORD
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nonn
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AUTHOR
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Frank Ruskey
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STATUS
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approved
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