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A184277 Table read by antidiagonals: T(n,k) = number of distinct n X k toroidal 0..3 arrays.

%I A184277 #31 Nov 30 2023 12:13:25
%S A184277 4,10,10,24,76,24,70,700,700,70,208,8296,29184,8296,208,700,104968,
%T A184277 1398500,1398500,104968,700,2344,1399176,71582944,268447936,71582944,
%U A184277 1399176,2344,8230,19175140,3817765120,54975633976,54975633976
%N A184277 Table read by antidiagonals: T(n,k) = number of distinct n X k toroidal 0..3 arrays.
%H A184277 Andrew Howroyd, <a href="/A184277/b184277.txt">Table of n, a(n) for n = 1..1275</a> (first 48 terms from R. H. Hardin)
%H A184277 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 A184277 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 A184277 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 A184277 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 pp. 3, 42.
%F A184277 T(n,k) = (1/(n*k)) * Sum_{c|n} Sum_{d|k} phi(c) * phi(d) * 4^(n*k/lcm(c,d)). - _Andrew Howroyd_, Sep 27 2017
%e A184277 Table starts
%e A184277       4        10           24             70            208            700
%e A184277      10        76          700           8296         104968        1399176
%e A184277      24       700        29184        1398500       71582944     3817765120
%e A184277      70      8296      1398500      268447936    54975633976 11728126132976
%e A184277     208    104968     71582944    54975633976 45035996274688
%e A184277     700   1399176   3817765120 11728126132976
%e A184277    2344  19175140 209430787824
%e A184277    8230 268447816
%e A184277   29144
%t A184277 T[n_, k_] := (1/(n*k))*Sum[Sum[EulerPhi[c]*EulerPhi[d]*4^(n*k/LCM[c, d]), {d, Divisors[k]}], {c, Divisors[n]}];
%t A184277 Table[T[n-k+1, k], {n, 1, 9}, {k, 1, n}] // Flatten (* _Jean-François Alcover_, Oct 31 2017, after _Andrew Howroyd_ *)
%o A184277 (PARI)
%o A184277 T(n, k) = (1/(n*k)) * sumdiv(n, c, sumdiv(k, d, eulerphi(c) * eulerphi(d) * 4^(n*k/lcm(c,d)))); \\ _Andrew Howroyd_, Sep 27 2017
%Y A184277 Columns 1-5 are A001868, A184273, A184274, A184275, A184276.
%Y A184277 Main diagonal is A184272.
%Y A184277 Cf. A184271, A184284.
%K A184277 nonn,tabl
%O A184277 1,1
%A A184277 _R. H. Hardin_, Jan 10 2011