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A307988 T(n, k) the number of A-polynomials in F_2^k[T] of degree n, array read by descending antidiagonals.

%I A307988 #17 May 25 2019 18:04:58
%S A307988 1,2,1,1,2,0,4,7,4,1,11,36,42,18,2,14,121,344,259,48,2,29,518,2750,
%T A307988 4068,1652,172,4,72,2059,21924,65461,52368,10962,588,9,127,8136,
%U A307988 174986,1048950,1677940,699288,74998,2034,14,242,32893,1398576,16778791,53686584,44738782,9587880,524475,7308,24
%N A307988 T(n, k) the number of A-polynomials in F_2^k[T] of degree n, array read by descending antidiagonals.
%H A307988 Alp Bassa, Ricardo Menares, <a href="https://arxiv.org/abs/1905.08345">Enumeration of a special class of irreducible polynomials in characteristic 2</a>, arXiv:1905.08345 [math.NT], 2019.
%H A307988 Harald Niederreiter, <a href="https://doi.org/10.1007/BF01810295">An enumeration formula for certain irreducible polynomials with an application to the construction of irreducible polynomials over the binary field</a>, Applicable Algebra in Engineering, Communication and Computing, vol.1, no.2, pp.119-124, (September-1990).
%F A307988 T(n, k) = Sum_{d|n} moebius(m/d)*q^(2^k*d) + 1 - alpha^(r*2^k*d) - alphabar^(r*2^k*d), where n = 2^k*m, m odd, alpha = (-1+sqrt(-7))/2 and alphabar = (-1-sqrt(-7))/2 is the conjugate of alpha.
%e A307988 Array begins:
%e A307988 1   2     1       4         11           14             29
%e A307988 1   2     7      36        121          518           2059
%e A307988 0   4    42     344       2750        21924         174986
%e A307988 1  18   259    4068      65461      1048950       16778791
%e A307988 2  48  1652   52368    1677940     53686584     1717985404
%e A307988 2 172 10962  699288   44738782   2863291620   183251786538
%e A307988 4 588 74998 9587880 1227132434 157072960476 20105353937606
%o A307988 (PARI) f(n) = 2 * real(((-1 + quadgen(-28)) / 2)^n);
%o A307988 a(n, r) = {my(k = valuation(n, 2), m = n/2^k, q = 2^r); sumdiv(m, d, moebius(m/d)*(q^(2^k*d)+1-f(r*2^k*d)))/(4*n);}
%Y A307988 Cf. A175390 (1st column).
%Y A307988 Cf. A002249 or A077021 (sequences related to alpha).
%K A307988 nonn,tabl
%O A307988 1,2
%A A307988 _Michel Marcus_, May 22 2019