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A284871 Array read by antidiagonals: T(n,k) = number of primitive (aperiodic) reversible strings of length n using a maximum of k different symbols.

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