%I #19 Jun 11 2017 20:55:00
%S 1,2,0,3,1,0,4,3,4,0,5,6,15,7,0,6,10,36,39,18,0,7,15,70,126,132,29,0,
%T 8,21,120,310,540,357,70,0,9,28,189,645,1620,2034,1131,126,0,10,36,
%U 280,1197,3990,7790,8316,3276,266,0
%N Array read by antidiagonals: T(n,k) = number of primitive (aperiodic) reversible strings of length n using a maximum of k different symbols.
%C A string and its reverse are considered to be equivalent.
%D M. R. Nester (1999). Mathematical investigations of some plant interaction designs. PhD Thesis. University of Queensland, Brisbane, Australia. [See A056391 for pdf file of Chap. 2]
%H Andrew Howroyd, <a href="/A284871/b284871.txt">Table of n, a(n) for n = 1..1275</a>
%F T(n, k) = Sum_{d | n} mu(n/d) * (k^n + k^(ceiling(n/2))) / 2.
%e Table starts:
%e 1 2 3 4 5 6 7 8 ...
%e 0 1 3 6 10 15 21 28 ...
%e 0 4 15 36 70 120 189 280 ...
%e 0 7 39 126 310 645 1197 2044 ...
%e 0 18 132 540 1620 3990 8568 16632 ...
%e 0 29 357 2034 7790 23295 58779 131012 ...
%e 0 70 1131 8316 39370 140610 412965 1050616 ...
%e 0 126 3276 32760 195300 839790 2882376 8388576 ...
%e ...
%t b[n_, k_] := (k^n + k^Ceiling[n/2])/2;
%t a[n_, k_] := DivisorSum[n, MoebiusMu[n/#] b[#, k]&];
%t Table[a[n-k+1, k], {n, 1, 10}, {k, n, 1, -1}] // Flatten (* _Jean-François Alcover_, Jun 05 2017, translated from PARI *)
%o (PARI)
%o b(n,k) = (k^n + k^(ceil(n/2))) / 2;
%o a(n,k) = sumdiv(n,d, moebius(n/d) * b(d,k));
%o for(n=1, 10, for(k=1, 10, print1( a(n,k),", ");); print(););
%Y Columns 2-6 are A045625, A056314, A056315, A056316, A056317.
%Y Cf. A277504, A143324.
%K nonn,tabl
%O 1,2
%A _Andrew Howroyd_, Apr 04 2017