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A284823 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).

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