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

Author

Gus Wiseman, Jan 17 2019

Keywords

• nonn

Comments

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

Examples

			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]
		

Links

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

Crossrefs

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

Programs

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}]

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

Extensions

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