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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. 1
 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. LINKS Alp Bassa, Ricardo Menares, Enumeration of a special class of irreducible polynomials in characteristic 2, arXiv:1905.08345 [math.NT], 2019. 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). 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 October 14 04:44 EDT 2019. Contains 327995 sequences. (Running on oeis4.)