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
Examples
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
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).
Programs
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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);}
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.