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A042980 Number of degree-n irreducible polynomials over GF(2) with trace = 0 and subtrace = 0. 8
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,6

LINKS

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

FORMULA

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)}.

MATHEMATICA

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-Fran├žois Alcover, Jun 28 2012, from formula *)

PROG

(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 */

CROSSREFS

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

KEYWORD

nonn,nice,easy

AUTHOR

Frank Ruskey

STATUS

approved

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Last modified September 21 11:46 EDT 2020. Contains 337268 sequences. (Running on oeis4.)