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A307988
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T(n, k) the number of A-polynomials in F_2^k[T] of degree n, array read by descending antidiagonals.
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0
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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
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OFFSET
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1,2
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LINKS
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FORMULA
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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.
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EXAMPLE
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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
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PROG
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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); }
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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