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A321720
Number of non-normal (0,1) semi-magic squares with sum of entries equal to n.
7
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.
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}]
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