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%I #32 Oct 28 2021 06:28:09
%S 8,36,36,176,1072,176,1044,43800,43800,1044,6560,2098720,14913536,
%T 2098720,6560,43800,107377488,5726645688,5726645688,107377488,43800,
%U 299600,5726689312,2345624810432,17592189193216,2345624810432,5726689312,299600
%N Table read by antidiagonals: T(n,k) = number of distinct n X k toroidal 0..7 arrays.
%H Alois P. Heinz, <a href="/A184294/b184294.txt">Antidiagonals n = 1..65, flattened</a> (first 8 antidiagonals from R. H. Hardin)
%H 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 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 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.
%F T(n,k) = (1/(n*k)) * Sum_{c|n} Sum_{d|k} phi(c) * phi(d) * 8^(n*k/lcm(c,d)). - _Andrew Howroyd_, Sep 27 2017
%e Table starts
%e 8 36 176 1044 6560 43800
%e 36 1072 43800 2098720 107377488 5726689312
%e 176 43800 14913536 5726645688 2345624810432
%e 1044 2098720 5726645688 17592189193216
%e 6560 107377488 2345624810432
%e 43800 5726689312
%e 299600
%p with(numtheory):
%p T:= (n, k)-> add(add(phi(c)*phi(d)*8^(n*k/ilcm(c, d)),
%p c=divisors(n)), d=divisors(k))/(n*k):
%p seq(seq(T(n, 1+d-n), n=1..d), d=1..8); # _Alois P. Heinz_, Aug 20 2017
%t T[n_, k_] := (1/(n*k))*Sum[Sum[EulerPhi[c]*EulerPhi[d]*8^(n*(k/LCM[c, d])), {d, Divisors[k]}], {c, Divisors[n]}]; Table[T[n - k + 1, k], {n, 1, 8}, {k, 1, n}] // Flatten (* _Jean-François Alcover_, Oct 30 2017, after _Alois P. Heinz_ *)
%o (PARI)
%o T(n, k) = (1/(n*k)) * sumdiv(n, c, sumdiv(k, d, eulerphi(c) * eulerphi(d) * 8^(n*k/lcm(c,d)))); \\ _Andrew Howroyd_, Sep 27 2017
%Y Columns 1-3 are A054627, A184292, A184293.
%Y Cf. A184271, A184284.
%K nonn,tabl
%O 1,1
%A _R. H. Hardin_, Jan 10 2011