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A042980
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Number of degree-n irreducible polynomials over GF(2) with trace = 0 and subtrace = 0.
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5
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1, 0, 0, 1, 1, 2, 5, 6, 15, 24, 45, 85, 155, 288, 550, 1008, 1935, 3626, 6885, 13107, 24940, 47616, 91225, 174590, 335626, 645120, 1242600, 2396745, 4627915, 8947294, 17318945, 33552384, 65076240, 126320640, 245424829, 477218560, 928638035, 1808400384, 3524082400, 6871921458, 13408691175, 26178823218
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OFFSET
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1,6
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REFERENCES
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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.
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LINKS
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Vincenzo Librandi, Table of n, a(n) for n = 1..1000
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{ L(n, k) : n+k = 2 mod 4}, where L(n, k) = Sum{ mu(d)*{n/d choose k/d}: d|gcd(n, k)}.
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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]==2, 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==2, 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. A042979-A042982.
Cf. A074027-A074030.
Sequence in context: A193405 A037079 A101325 * A048290 A029939 A082198
Adjacent sequences: A042977 A042978 A042979 * A042981 A042982 A042983
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KEYWORD
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nonn,nice,easy
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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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