A184331 - OEIS

%I #30 Oct 28 2021 06:28:04

%S 7,28,28,119,637,119,616,19684,19684,616,3367,721525,4484039,721525,

%T 3367,19684,28249228,1153450872,1153450872,28249228,19684,117655,

%U 1153470437,316504102999,2077059243301,316504102999,1153470437,117655,720916

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

%H 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 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) * 7^(n*k/lcm(c,d)). - _Andrew Howroyd_, Sep 27 2017

%e Table starts

%e        7         28          119           616         3367          19684

%e       28        637        19684        721525     28249228     1153470437

%e      119      19684      4484039    1153450872 316504102999 90467424400444

%e      616     721525   1153450872 2077059243301

%e     3367   28249228 316504102999

%e    19684 1153470437

%e   117655

%t 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 (PARI)

%o 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 Columns 1-3 are A054626, A184329, A184330.

%Y Cf. A184271, A184284.

%K nonn,tabl

%O 1,1

%A _R. H. Hardin_, Jan 11 2011