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A184294 Table read by antidiagonals: T(n,k) = number of distinct n X k toroidal 0..7 arrays.

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