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A184288 Table read by antidiagonals: T(n,k) = number of distinct n X k toroidal 0..4 arrays.

%I A184288 #23 Oct 27 2021 11:44:06
%S A184288 5,15,15,45,175,45,165,2635,2635,165,629,49075,217125,49075,629,2635,
%T A184288 976887,20346485,20346485,976887,2635,11165,20349075,2034505661,
%U A184288 9536816875,2034505661,20349075,11165,48915,435970995,211927741375
%N A184288 Table read by antidiagonals: T(n,k) = number of distinct n X k toroidal 0..4 arrays.
%H A184288 Andrew Howroyd, <a href="/A184288/b184288.txt">Table of n, a(n) for n = 1..1275</a> (first 39 terms from R. H. Hardin)
%H A184288 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 A184288 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 A184288 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 A184288 T(n,k) = (1/(n*k)) * Sum_{c|n} Sum_{d|k} phi(c) * phi(d) * 5^(n*k/lcm(c,d)). - _Andrew Howroyd_, Sep 27 2017
%e A184288 Table starts
%e A184288       5        15           45           165           629         2635
%e A184288      15       175         2635         49075        976887     20349075
%e A184288      45      2635       217125      20346485    2034505661 211927741375
%e A184288     165     49075     20346485    9536816875 4768372070757
%e A184288     629    976887   2034505661 4768372070757
%e A184288    2635  20349075 211927741375
%e A184288   11165 435970995
%e A184288   48915
%t A184288 T[n_, k_] := (1/(n*k))*Sum[Sum[EulerPhi[c] * EulerPhi[d] * 5^(n*k/LCM[c, d]), {d, Divisors[k]}], {c, Divisors[n]}];
%t A184288 Table[T[n - k + 1, k], {n, 1, 9}, {k, 1, n}] // Flatten (* _Jean-François Alcover_, Oct 31 2017, after _Andrew Howroyd_ *)
%o A184288 (PARI)
%o A184288 T(n, k) = (1/(n*k)) * sumdiv(n, c, sumdiv(k, d, eulerphi(c) * eulerphi(d) * 5^(n*k/lcm(c,d)))); \\ _Andrew Howroyd_, Sep 27 2017
%Y A184288 Columns 1-4 are A001869, A184286, A184287, A184288.
%Y A184288 Cf. A184271, A184284.
%K A184288 nonn,tabl
%O A184288 1,1
%A A184288 _R. H. Hardin_, Jan 10 2011