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A184291 Table read by antidiagonals: T(n,k) = number of distinct n X k toroidal 0..5 arrays.

%I A184291 #28 Oct 27 2021 11:33:02
%S A184291 6,21,21,76,351,76,336,7826,7826,336,1560,210456,1119936,210456,1560,
%T A184291 7826,6047412,181402676,181402676,6047412,7826,39996,181410426,
%U A184291 31345666736,176319685116,31345666736,181410426,39996,210126,5597460306
%N A184291 Table read by antidiagonals: T(n,k) = number of distinct n X k toroidal 0..5 arrays.
%H A184291 Andrew Howroyd, <a href="/A184291/b184291.txt">Table of n, a(n) for n = 1..1275</a> (first 31 terms from R. H. Hardin)
%H A184291 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 A184291 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 A184291 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 A184291 T(n,k) = (1/(n*k)) * Sum_{c|n} Sum_{d|k} phi(c) * phi(d) * 6^(n*k/lcm(c,d)). - _Andrew Howroyd_, Sep 27 2017
%e A184291 Table starts
%e A184291       6        21          76          336        1560          7826      39996
%e A184291      21       351        7826       210456     6047412     181410426 5597460306
%e A184291      76      7826     1119936    181402676 31345666736 5642220395616
%e A184291     336    210456   181402676 176319685116
%e A184291    1560   6047412 31345666736
%e A184291    7826 181410426
%e A184291   39996
%t A184291 T[n_, k_] := (1/(n*k))*Sum[Sum[EulerPhi[c]*EulerPhi[d]*6^(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 _Andrew Howroyd_ *)
%o A184291 (PARI)
%o A184291 T(n, k) = (1/(n*k)) * sumdiv(n, c, sumdiv(k, d, eulerphi(c) * eulerphi(d) * 6^(n*k/lcm(c,d)))); \\ _Andrew Howroyd_, Sep 27 2017
%Y A184291 Columns 1-3 are A054625, A184289, A184290.
%Y A184291 Cf. A184271, A184284.
%K A184291 nonn,tabl
%O A184291 1,1
%A A184291 _R. H. Hardin_, Jan 10 2011