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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. 2
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 (list; graph; refs; listen; history; text; internal format)
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 (zeroes not shown):

  [3  ] [2 1] [1 2] [  3]

  [  3] [1 2] [2 1] [3  ]

The a(6,3) = 21 semi-magic squares (zeroes 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: A225766 A249831 A304527 * A154744 A285038 A243145

Adjacent sequences:  A321722 A321723 A321724 * A321726 A321727 A321728

KEYWORD

nonn,more,tabf

AUTHOR

Gus Wiseman, Nov 18 2018

EXTENSIONS

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

STATUS

approved

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Last modified February 24 01:14 EST 2020. Contains 332195 sequences. (Running on oeis4.)