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

Author

Gus Wiseman, Nov 18 2018

Keywords

• nonn

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

Crossrefs

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

Cf. A321717, A321719, A321722, A321723.

Programs

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}]

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

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