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A175390 Number of irreducible binary polynomials sum(j=0..n, c(j)*x^j) with c(1)=c(n-1)=1. 0
1, 1, 0, 1, 2, 2, 4, 9, 14, 24, 48, 86, 154, 294, 550, 1017, 1926, 3654, 6888, 13092, 24998, 47658, 91124, 174822, 335588, 645120, 1242822, 2396970, 4627850, 8947756, 17319148, 33553881, 65074406, 126324420, 245426486, 477215270, 928645186 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,5

COMMENTS

Binary polynomial means polynomial over GF(2).

A formula for the enumeration is given in Niederreiter's paper, see the pari/gp code.

a(n)>0 for n>3.

REFERENCES

Harald Niederreiter: An enumeration formula for certain irreducible polynomials with an application to the construction of irreducible polynomials over the binary field, Applicable Algebra in Engineering, Communication and Computing, vol.1, no.2, pp.119-124, (September-1990)

LINKS

Table of n, a(n) for n=1..37.

EXAMPLE

The only irreducible binary polynomial of degree 2 is x^2+x+1 and it has the required property, so a(2)=1. The only polynomials of degree 3 with c(1)=c(2)=1 are x^3+x^2+x and x^3+x^2+x+1; neither is irreducible, so a(3)=0.

PROG

(PARI)

A(n) = {

my( h, m, ret );

if ( n==1, return(1) );

h = valuation(n, 2); /* largest power of 2 dividing n */

m = n/2^h; /* odd part of n */

if ( m == 1, /* power of two */

  ret = (2^n+1)/(4*n) - 1/(2^(n+1)*n) * sum(j=0, n/2, (-1)^j*binomial(n, 2*j)*7^j);

, /* else */

  ret = 1/(4*n)*sumdiv(m, d, moebius(m/d) *(2^(2^h*d) - 2^(1-2^h*d)*sum(j=0, floor(2^(h-1)*d), (-1)^(2^h*d+j) * binomial(2^h*d, 2*j)*7^j) ) );

);

return( ret );

}

vector(50, n, A(n))

CROSSREFS

Sequence in context: A257515 A105152 A066346 * A054233 A054231 A054230

Adjacent sequences:  A175387 A175388 A175389 * A175391 A175392 A175393

KEYWORD

nonn

AUTHOR

Joerg Arndt, Apr 27 2010

EXTENSIONS

Edited by Franklin T. Adams-Watters, May 12 2010

STATUS

approved

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Last modified March 24 22:17 EDT 2017. Contains 284035 sequences.