A323523 Number of positive integer square matrices with entries summing to n and equal row and column sums.

1, 1, 1, 1, 2, 1, 3, 1, 4, 2, 5, 1, 12, 1, 7, 22, 9, 1, 64, 1, 34, 121, 11, 1, 525, 2, 13, 407, 2022, 1, 801, 1, 10163, 1036, 17, 6211, 41735, 1, 19, 2212, 285784, 1, 3822, 1, 381446, 2229142, 23, 1, 1189540, 2, 22069276, 7261, 2309410, 1, 20943183, 164176641

Offset

0,5

Comments

Also the number of non-normal semi-magic squares with positive integer entries summing to n.

Links

Andrew Howroyd, Table of n, a(n) for n = 0..200 (terms 0..59 from Chai Wah Wu)

Formula

a(p) = 1 and a(p^2) = 2 for p prime (see comment in A323349). - Chai Wah Wu, Jan 20 2019

a(n) = Sum_{d|n, d<=n/d} A257493(d, n/d-d) for n > 0. - Andrew Howroyd, Apr 10 2020

Example

The a(12) = 12 matrices:
  [12]
.
  [1 5] [5 1] [2 4] [4 2] [3 3]
  [5 1] [1 5] [4 2] [2 4] [3 3]
.
  [1 1 2] [1 1 2] [1 2 1] [1 2 1] [2 1 1] [2 1 1]
  [1 2 1] [2 1 1] [1 1 2] [2 1 1] [1 1 2] [1 2 1]
  [2 1 1] [1 2 1] [2 1 1] [1 1 2] [1 2 1] [1 1 2]

Mathematica

primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
facs[n_]:=If[n<=1, {{}}, Join@@Table[Map[Prepend[#, d]&, Select[facs[n/d], Min@@#>=d&]], {d, Rest[Divisors[n]]}]];
ptnsqrs[n_]:=Union@@Permutations/@Select[Union@@(Tuples[Permutations/@#]&/@Map[primeMS, facs[n], {2}]), And[SameQ@@Length/@#, Length[#]==0||Length[#]==Length[First[#]]]&];
Table[Sum[Length[Select[ptnsqrs[Times@@Prime/@y], And[SameQ@@Total/@#, SameQ@@Total/@Transpose[#]]&]], {y, IntegerPartitions[n]}], {n, 10}]

Crossrefs

Cf. A000290, A006052, A007016, A103198, A120732, A257493, A321719, A321722, A323349, A323523, A323524, A323529.

Sequence in context: A143862 A115118 A115121 * A124072 A189357 A100053

Adjacent sequences: A323520 A323521 A323522 * A323524 A323525 A323526

Keyword

nonn

Author

Gus Wiseman, Jan 17 2019

Extensions

a(16)-a(55) from Chai Wah Wu, Jan 20 2019

Status

approved