A321725 Irregular triangle read by rows where T(n,d) is the number of d X d non-normal semi-magic squares with sum of all entries equal to n.

1, 1, 2, 1, 6, 1, 3, 24, 1, 120, 1, 4, 21, 720, 1, 5040, 1, 5, 282, 40320, 1, 55, 362880, 1, 6, 6210, 3628800, 1, 39916800, 1, 7, 120, 2008, 202410, 479001600, 1, 6227020800, 1, 8, 9135630, 87178291200, 1, 231, 153040, 1307674368000, 1, 9, 10147

Offset

1,3

Comments

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

Links

Chai Wah Wu, Table of n, a(n) for n = 1..60

Wikipedia, Magic square

Index entries for sequences related to magic squares

Formula

T(n,n) = n!. Sum_d T(n,d) = A321719(n). - Chai Wah Wu, Jan 15 2019

Example

Triangle begins:
   1
   1   2
   1   6
   1   3  24
   1 120
   1   4  21 720
The a(6,2) = 4 semi-magic squares (zeros not shown):
  [3  ] [2 1] [1 2] [  3]
  [  3] [1 2] [2 1] [3  ]
The a(6,3) = 21 semi-magic squares (zeros not shown):
  [2    ] [2    ] [2    ] [1 1  ] [1 1  ] [1 1  ] [1 1  ]
  [  2  ] [  1 1] [    2] [1 1  ] [1   1] [  1 1] [    2]
  [    2] [  1 1] [  2  ] [    2] [  1 1] [1   1] [1 1  ]
.
  [1   1] [1   1] [1   1] [1   1] [  2  ] [  2  ] [  2  ]
  [1 1  ] [1   1] [  2  ] [  1 1] [2    ] [1   1] [    2]
  [  1 1] [  2  ] [1   1] [1 1  ] [    2] [1   1] [2    ]
.
  [  1 1] [  1 1] [  1 1] [  1 1] [    2] [    2] [    2]
  [2    ] [1 1  ] [1   1] [  1 1] [2    ] [1 1  ] [  2  ]
  [  1 1] [1   1] [1 1  ] [2    ] [  2  ] [1 1  ] [2    ]

Mathematica

prs2mat[prs_]:=Table[Count[prs, {i, j}], {i, Union[First/@prs]}, {j, Union[Last/@prs]}];
multsubs[set_, k_]:=If[k==0, {{}}, Join@@Table[Prepend[#, set[[i]]]&/@multsubs[Drop[set, i-1], k-1], {i, Length[set]}]];
Table[Length[Select[multsubs[Tuples[Range[n], 2], n], And[Union[First/@#]==Range[k]==Union[Last/@#], SameQ@@Total/@prs2mat[#], SameQ@@Total/@Transpose[prs2mat[#]]]&]], {n, 5}, {k, Divisors[n]}]

Crossrefs

Cf. A006052, A007016, A120732, A319056, A319616.

Cf. A321718, A321719, A321721, A321722, A321724.

Sequence in context: A334085 A249831 A304527 * A154744 A285038 A243145

Adjacent sequences: A321722 A321723 A321724 * A321726 A321727 A321728

Keyword

nonn,tabf

Author

Gus Wiseman, Nov 18 2018

Extensions

a(15)-a(48) from Chai Wah Wu, Jan 15 2019

Status

approved