A321720 Number of non-normal (0,1) semi-magic squares with sum of entries equal to n.

1, 1, 2, 6, 25, 120, 726, 5040, 40410, 362881, 3630840, 39916800, 479069574, 6227020800, 87181402140, 1307674370040, 20922977418841, 355687428096000, 6402388104196400, 121645100408832000, 2432903379962038320, 51090942171778378800, 1124000886592995642000, 25852016738884976640000

Offset

0,3

Comments

A non-normal semi-magic square is a nonnegative integer matrix with row sums and column sums all equal to d, for some d|n.

Links

Andrew Howroyd, Table of n, a(n) for n = 0..180

Wikipedia, Magic square

Index entries for sequences related to magic squares

Formula

a(p) = p! for p prime as the squares are all permutation matrices of order p and a(n) >= n! for n > 1 (see comments in A321717 and A321719). - Chai Wah Wu, Jan 13 2019

a(n) = Sum_{d|n, d<=n/d} A008300(n/d, d) for n > 0. - Andrew Howroyd, Apr 11 2020

Mathematica

prs2mat[prs_]:=Table[Count[prs, {i, j}], {i, Union[First/@prs]}, {j, Union[Last/@prs]}];
Table[Length[Select[Subsets[Tuples[Range[n], 2], {n}], And[Union[First/@#]==Union[Last/@#]==Range[Max@@First/@#], SameQ@@Total/@prs2mat[#], SameQ@@Total/@Transpose[prs2mat[#]]]&]], {n, 5}]

Crossrefs

Cf. A006052, A007016, A057151, A068313, A008300, A101370, A104602, A120732, A271103, A319056, A319616.

Cf. A321717, A321719, A321722, A321723.

Sequence in context: A006965 A202705 A058801 * A358499 A357949 A321186

Adjacent sequences: A321717 A321718 A321719 * A321721 A321722 A321723

Keyword

nonn

Author

Gus Wiseman, Nov 18 2018

Extensions

a(7) from Chai Wah Wu, Jan 13 2019

a(8)-a(15) from Chai Wah Wu, Jan 14 2019

a(16)-a(21) from Chai Wah Wu, Jan 16 2019

Terms a(22) and beyond from Andrew Howroyd, Apr 11 2020

Status

approved