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

1, 2, 1, 1, 2, 0, 4, 7, 4, 1, 11, 36, 42, 18, 2, 14, 121, 344, 259, 48, 2, 29, 518, 2750, 4068, 1652, 172, 4, 72, 2059, 21924, 65461, 52368, 10962, 588, 9, 127, 8136, 174986, 1048950, 1677940, 699288, 74998, 2034, 14, 242, 32893, 1398576, 16778791, 53686584, 44738782, 9587880, 524475, 7308, 24

Offset

1,2

Links

Table of n, a(n) for n=1..55.

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).

Formula

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.

Example

Array begins:
1   2     1       4         11           14             29
1   2     7      36        121          518           2059
0   4    42     344       2750        21924         174986
1  18   259    4068      65461      1048950       16778791
2  48  1652   52368    1677940     53686584     1717985404
2 172 10962  699288   44738782   2863291620   183251786538
4 588 74998 9587880 1227132434 157072960476 20105353937606

Prog

(PARI) f(n) = 2 * real(((-1 + quadgen(-28)) / 2)^n);
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); }

Crossrefs

Cf. A175390 (1st column).

Cf. A002249 or A077021 (sequences related to alpha).

Sequence in context: A358338 A244658 A117586 * A268917 A176811 A057594

Adjacent sequences: A307985 A307986 A307987 * A307989 A307990 A307991

Keyword

nonn,tabl

Author

Michel Marcus, May 22 2019

Status

approved