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%I #20 Apr 24 2020 18:38:00
%S 1,2,0,3,0,0,4,0,2,0,5,0,6,2,0,6,0,12,6,6,0,7,0,20,12,24,4,0,8,0,30,
%T 20,60,18,14,0,9,0,42,30,120,48,78,12,0,10,0,56,42,210,100,252,72,28,
%U 0,11,0,72,56,336,180,620,240,234,24,0,12,0,90,72,504,294,1290,600,1008,216,62
%N Array read by antidiagonals: T(n,k) = number of primitive (aperiodic) palindromes of length n using a maximum of k different symbols (n >= 1, k >= 1).
%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="/A284823/b284823.txt">Table of n, a(n) for n = 1..1275</a>
%F T(n,k) = Sum_{d | n} mu(n/d) * k^(ceiling(d/2)).
%e Table starts:
%e 1 2 3 4 5 6 7 8 9 10 ...
%e 0 0 0 0 0 0 0 0 0 0 ...
%e 0 2 6 12 20 30 42 56 72 90 ...
%e 0 2 6 12 20 30 42 56 72 90 ...
%e 0 6 24 60 120 210 336 504 720 990 ...
%e 0 4 18 48 100 180 294 448 648 900 ...
%e 0 14 78 252 620 1290 2394 4088 6552 9990 ...
%e 0 12 72 240 600 1260 2352 4032 6480 9900 ...
%e 0 28 234 1008 3100 7740 16758 32704 58968 99900 ...
%e 0 24 216 960 3000 7560 16464 32256 58320 99000 ...
%e ...
%e Row 4 includes palindromes of the form abba but excludes those of the form aaaa, so T(4,k) is k*(k-1).
%e Row 6 includes palindromes of the forms aabbaa, abbbba, abccba but excludes those of the forms aaaaaa, abaaba, so T(6,k) is 2*k*(k-1) + k*(k-1)*(k-2).
%t T[n_, k_] := DivisorSum[n, MoebiusMu[n/#]*k^Ceiling[#/2]&]; Table[T[n-k+1, k], {n, 1, 12}, {k, n, 1, -1}] // Flatten (* _Jean-François Alcover_, Jun 05 2017 *)
%o (PARI)
%o a(n,k) = sumdiv(n, d, moebius(n/d) * k^(ceil(d/2)));
%o for(n=1, 10, for(k=1, 10, print1( a(n,k),", ");); print();)
%Y Columns 2-6 are A056458, A056459, A056460, A056461, A056462.
%Y Rows 5-10 are A007531(k+1), A045991, A058895, A047928(k-1), A135497, A133754.
%Y Cf. A284826, A284841.
%K nonn,tabl
%O 1,2
%A _Andrew Howroyd_, Apr 03 2017
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