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A284856 Array read by antidiagonals: T(n,k) = number of aperiodic necklaces (Lyndon words) with n beads and k colors that are the same when turned over.

%I A284856 #12 Jun 06 2017 11:55:00
%S A284856 1,2,0,3,1,0,4,3,2,0,5,6,6,3,0,6,10,12,12,6,0,7,15,20,30,24,7,0,8,21,
%T A284856 30,60,60,42,14,0,9,28,42,105,120,138,78,18,0,10,36,56,168,210,340,
%U A284856 252,144,28,0,11,45,72,252,336,705,620,600,234,39,0
%N A284856 Array read by antidiagonals: T(n,k) = number of aperiodic necklaces (Lyndon words) with n beads and k colors that are the same when turned over.
%C A284856 Number of primitive (period n) periodic palindromes of length n using a maximum of k different symbols.
%D A284856 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 A284856 Andrew Howroyd, <a href="/A284856/b284856.txt">Table of n, a(n) for n = 1..1275</a>
%F A284856 T(n, k) = Sum_{d | n} mu(n/d) * A284855(d, k).
%e A284856 Table starts:
%e A284856 1  2   3    4    5     6     7      8      9     10 ...
%e A284856 0  1   3    6   10    15    21     28     36     45 ...
%e A284856 0  2   6   12   20    30    42     56     72     90 ...
%e A284856 0  3  12   30   60   105   168    252    360    495 ...
%e A284856 0  6  24   60  120   210   336    504    720    990 ...
%e A284856 0  7  42  138  340   705  1302   2212   3528   5355 ...
%e A284856 0 14  78  252  620  1290  2394   4088   6552   9990 ...
%e A284856 0 18 144  600 1800  4410  9408  18144  32400  54450 ...
%e A284856 0 28 234 1008 3100  7740 16758  32704  58968  99900 ...
%e A284856 0 39 456 2490 9240 26985 66864 146916 294480 548955 ...
%e A284856 ...
%t A284856 b[d_, k_] := If[EvenQ[d], (k^(d/2) + k^(d/2 + 1))/2, k^((d + 1)/2)];
%t A284856 a[n_, k_] := DivisorSum[n, MoebiusMu[n/#] b[#, k] &];
%t A284856 Table[a[n - k + 1, k], {n, 1, 11}, {k, n, 1, -1}] // Flatten (* _Jean-François Alcover_, Jun 06 2017, translated from PARI *)
%o A284856 (PARI)
%o A284856 b(d,k) = if(d % 2 == 0, (k^(d/2) + k^(d/2+1))/2, k^((d+1)/2));
%o A284856 a(n,k) = sumdiv(n,d, moebius(n/d) * b(d,k));
%o A284856 for(n=1, 10, for(k=1, 10, print1( a(n,k),", ");); print(););
%Y A284856 Columns 2-6 are: A056493, A056494, A056495, A056496, A056497.
%K A284856 nonn,tabl
%O A284856 1,2
%A A284856 _Andrew Howroyd_, Apr 04 2017