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A098691 Array T(q,n) by antidiagonals: number of self-reciprocal polynomials of degree 2*n over GF(q) (for q >= 2 and n >= 1).

%I A098691 #23 Aug 22 2019 22:23:09
%S A098691 1,1,1,2,2,1,2,4,4,2,3,6,10,10,3,3,9,20,32,24,5,4,12,35,78,102,60,9,4,
%T A098691 16,56,162,312,340,156,16,5,20,84,300,777,1300,1170,410,28,5,25,120,
%U A098691 512,1680,3885,5580,4096,1092,51,6,30,165,820,3276,9800,19995,24414
%N A098691 Array T(q,n) by antidiagonals: number of self-reciprocal polynomials of degree 2*n over GF(q) (for q >= 2 and n >= 1).
%C A098691 Also, number of self-complementary primitive necklaces of length n in q colors.
%H A098691 Andrew Howroyd, <a href="/A098691/b098691.txt">Table of n, a(n) for n = 2..1276</a>
%H A098691 H. Meyn and W. Götz, <a href="http://www.mat.univie.ac.at/~slc/opapers/s21meyn.html">Self-reciprocal polynomials over finite fields</a>, Séminaire Lotharingien de Combinatoire, B21d (1989), 8 pp.
%H A098691 R. L. Miller, <a href="http://dx.doi.org/10.1016/0012-365X(78)90043-2">Necklaces, symmetries and self-reciprocal polynomials</a>, Discr. Math. 22 (1978), 25-33.
%F A098691 T(q, n) = (q^n-1)/(2*n) for q odd and n=2^s; otherwise Sum_{d|n, d odd} mu(d)*q^(n/d) / (2*n).
%F A098691 T(q, n) = Sum_{d|n, d odd} mu(d) * (q^(n/d) - (q mod 2)) / (2*n). - _Andrew Howroyd_, Aug 21 2019
%e A098691   [q=2]: 1,  1,  1,   2,    3,    5,     9,     16, ...
%e A098691   [q=3]: 1,  2,  4,  10,   24,   60,   156,    410, ...
%e A098691   [q=4]: 2,  4, 10,  32,  102,  340,  1170,   4096, ...
%e A098691   [q=5]: 2,  6, 20,  78,  312, 1300,  5580,  24414, ...
%e A098691   [q=6]: 3,  9, 35, 162,  777, 3885, 19995, 104976, ...
%e A098691   [q=7]: 3, 12, 56, 300, 1680, 9800, 58824, 360300, ...
%e A098691   ...
%o A098691 (PARI) T(q,n) = sumdiv(n, d, if(d%2, moebius(d) * (q^(n/d)-q%2), 0)) / (2*n); \\ _Andrew Howroyd_, Aug 21 2019
%o A098691 (PARI) T(q,n) = {if(q%2 && n == 2^logint(n,2), q^n-1, sumdiv(n, d, if(d%2, moebius(d)*q^(n/d)))) / (2*n)} \\ _Andrew Howroyd_, Aug 22 2019
%Y A098691 Rows are A000048 (q=2), A006575 (q=3).
%Y A098691 Columns 1-4 are A004526, A002620, A000292, 2*A011863.
%Y A098691 Main diagonal is in A098692.
%K A098691 nonn,tabl
%O A098691 2,4
%A A098691 _Ralf Stephan_, Sep 21 2004