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A184284 Table read by antidiagonals: T(n,k) = number of distinct n X k toroidal 0..2 arrays.

%I A184284 #30 Nov 30 2023 12:13:04
%S A184284 3,6,6,11,27,11,24,130,130,24,51,855,2211,855,51,130,5934,44368,44368,
%T A184284 5934,130,315,44487,956635,2691711,956635,44487,315,834,341802,
%U A184284 21524790,174342216,174342216,21524790,341802,834,2195,2691675,498112275
%N A184284 Table read by antidiagonals: T(n,k) = number of distinct n X k toroidal 0..2 arrays.
%H A184284 Alois P. Heinz, <a href="/A184284/b184284.txt">antidiagonals n = 1..50, flattened</a> (first 58 terms from R. H. Hardin)
%H A184284 S. N. Ethier, <a href="http://arxiv.org/abs/1301.2352">Counting toroidal binary arrays</a>, arXiv:1301.2352v1 [math.CO], Jan 10, 2013.
%H A184284 S. N. Ethier and Jiyeon Lee, <a href="http://arxiv.org/abs/1502.03792">Counting toroidal binary arrays, II</a>, arXiv:1502.03792v1 [math.CO], Feb 12, 2015.
%H A184284 Veronika Irvine, <a href="http://hdl.handle.net/1828/7495">Lace Tessellations: A mathematical model for bobbin lace and an exhaustive combinatorial search for patterns</a>, PhD Dissertation, University of Victoria, 2016.
%H A184284 Peter Kagey and William Keehn, <a href="https://arxiv.org/abs/2311.13072">Counting tilings of the n X m grid, cylinder, and torus</a>, arXiv:2311.13072 [math.CO], 2023. See p. 3.
%F A184284 T(n,k) = (1/(n*k)) * Sum_{c|n} Sum_{d|k} phi(c) * phi(d) * 3^(n*k/lcm(c,d)). - _Andrew Howroyd_, Sep 27 2017
%e A184284 Table starts
%e A184284      3        6          11           24            51           130
%e A184284      6       27         130          855          5934         44487
%e A184284     11      130        2211        44368        956635      21524790
%e A184284     24      855       44368      2691711     174342216   11767964475
%e A184284     51     5934      956635    174342216   33891544611 6863038218842
%e A184284    130    44487    21524790  11767964475 6863038218842
%e A184284    315   341802   498112275 817028472960
%e A184284    834  2691675 11767920118
%e A184284   2195 21524542
%e A184284   5934
%t A184284 T[n_, k_] := (1/(n*k))*Sum[EulerPhi[c]*EulerPhi[d]*3^(n*k/LCM[c, d]), {c, Divisors[n]}, {d, Divisors[k]}]; Table[T[n-k+1, k], {n, 1, 9}, {k, 1, n}] // Flatten (*_Jean-François Alcover_, Oct 07 2017, after _Andrew Howroyd_ *)
%o A184284 (PARI)
%o A184284 T(n, k) = (1/(n*k)) * sumdiv(n, c, sumdiv(k, d, eulerphi(c) * eulerphi(d) * 3^(n*k/lcm(c,d)))); \\ _Andrew Howroyd_, Sep 27 2017
%Y A184284 Columns 1-6 are A001867, A184279, A184280, A184281, A184282, A184283.
%Y A184284 Main diagonal is A184278.
%Y A184284 Cf. A184271, A184277, A184288, A184291, A184331, A184294 (0..1, 0..3 etc.).
%K A184284 nonn,tabl
%O A184284 1,1
%A A184284 _R. H. Hardin_, Jan 10 2011