A184288 - OEIS

A184288

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

%I #23 Oct 27 2021 11:44:06

%S 5,15,15,45,175,45,165,2635,2635,165,629,49075,217125,49075,629,2635,

%T 976887,20346485,20346485,976887,2635,11165,20349075,2034505661,

%U 9536816875,2034505661,20349075,11165,48915,435970995,211927741375

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

%H 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 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) * 5^(n*k/lcm(c,d)). - _Andrew Howroyd_, Sep 27 2017

%e Table starts

%e 5 15 45 165 629 2635

%e 15 175 2635 49075 976887 20349075

%e 45 2635 217125 20346485 2034505661 211927741375

%e 165 49075 20346485 9536816875 4768372070757

%e 629 976887 2034505661 4768372070757

%e 2635 20349075 211927741375

%e 11165 435970995

%e 48915

%t 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 Table[T[n - k + 1, k], {n, 1, 9}, {k, 1, n}] // Flatten (* _Jean-François Alcover_, Oct 31 2017, after _Andrew Howroyd_ *)

%o (PARI)

%o 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 Columns 1-4 are A001869, A184286, A184287, A184288.

%Y Cf. A184271, A184284.

%K nonn,tabl

%O 1,1

%A _R. H. Hardin_, Jan 10 2011