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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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8
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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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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_{ L(n, k) : n+k = 2 mod 4}, where L(n, k) = Sum_{ mu(d)*binomial(n/d, 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, A042981, A042982.
Cf. A074027, A074028, A074029, A074030.
Sequence in context: A193405 A037079 A101325 * A048290 A306885 A029939
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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