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A184331 Table read by antidiagonals: T(n,k) = number of distinct n X k toroidal 0..6 arrays.

%I A184331 #30 Oct 28 2021 06:28:04
%S A184331 7,28,28,119,637,119,616,19684,19684,616,3367,721525,4484039,721525,
%T A184331 3367,19684,28249228,1153450872,1153450872,28249228,19684,117655,
%U A184331 1153470437,316504102999,2077059243301,316504102999,1153470437,117655,720916
%N A184331 Table read by antidiagonals: T(n,k) = number of distinct n X k toroidal 0..6 arrays.
%H A184331 Andrew Howroyd, <a href="/A184331/b184331.txt">Table of n, a(n) for n = 1..1275</a> (first 31 terms from R. H. Hardin)
%H A184331 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 A184331 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 A184331 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 A184331 T(n,k) = (1/(n*k)) * Sum_{c|n} Sum_{d|k} phi(c) * phi(d) * 7^(n*k/lcm(c,d)). - _Andrew Howroyd_, Sep 27 2017
%e A184331 Table starts
%e A184331        7         28          119           616         3367          19684
%e A184331       28        637        19684        721525     28249228     1153470437
%e A184331      119      19684      4484039    1153450872 316504102999 90467424400444
%e A184331      616     721525   1153450872 2077059243301
%e A184331     3367   28249228 316504102999
%e A184331    19684 1153470437
%e A184331   117655
%t A184331 T[n_, k_] := (1/(n*k))*Sum[Sum[EulerPhi[c]*EulerPhi[d]*7^(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 A184331 (PARI)
%o A184331 T(n, k) = (1/(n*k)) * sumdiv(n, c, sumdiv(k, d, eulerphi(c) * eulerphi(d) * 7^(n*k/lcm(c,d)))); \\ _Andrew Howroyd_, Sep 27 2017
%Y A184331 Columns 1-3 are A054626, A184329, A184330.
%Y A184331 Cf. A184271, A184284.
%K A184331 nonn,tabl
%O A184331 1,1
%A A184331 _R. H. Hardin_, Jan 11 2011