A321725 Irregular triangle read by rows where T(n,k) is the number of d X d non-normal semi-magic squares with d = A027750(n,k) and 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

Author

Gus Wiseman, Nov 18 2018

Keywords

• nonn
• tabf

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.

Examples

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

Links

• Chai Wah Wu, Table of n, a(n) for n = 1..60
• Wikipedia, Magic square
• Index entries for sequences related to magic squares

Crossrefs

Cf. A006052, A007016, A027750, A120732, A319056, A319616.

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

Programs

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]}]

Formula

T(n, A000005(n)) = n!. Sum_k T(n,k) = A321719(n). - Chai Wah Wu, Jan 15 2019

Extensions

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

Edited by Peter Munn, Mar 05 2025