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%I #10 Apr 03 2024 05:14:05
%S 1,1,3,11,53,317,2293,19435,188851,2068417,25203807,338117445,
%T 4951449055,78589443061,1343810727205,24626270763109,481489261372381,
%U 10004230113283129,220125503239710879,5113204953106107087,125037079246130168973
%N Number of nonnegative integer square matrices with sum of entries equal to n, no zero rows or columns, and the same row sums as column sums.
%H Ludovic Schwob, <a href="/A321732/b321732.txt">Table of n, a(n) for n = 0..30</a>
%e The a(3) = 11 matrices:
%e [3]
%e .
%e [2 0] [1 1] [1 0] [0 1]
%e [0 1] [1 0] [0 2] [1 1]
%e .
%e [1 0 0] [1 0 0] [0 1 0] [0 1 0] [0 0 1] [0 0 1]
%e [0 1 0] [0 0 1] [1 0 0] [0 0 1] [1 0 0] [0 1 0]
%e [0 0 1] [0 1 0] [0 0 1] [1 0 0] [0 1 0] [1 0 0]
%t prs2mat[prs_]:=Table[Count[prs,{i,j}],{i,Union[First/@prs]},{j,Union[Last/@prs]}];
%t multsubs[set_,k_]:=If[k==0,{{}},Join@@Table[Prepend[#,set[[i]]]&/@multsubs[Drop[set,i-1],k-1],{i,Length[set]}]];
%t Table[Length[Select[multsubs[Tuples[Range[n],2],n],And[Union[First/@#]==Range[Max@@First/@#]==Union[Last/@#],Total/@prs2mat[#]==Total/@Transpose[prs2mat[#]]]&]],{n,5}]
%Y Cf. A000700, A000701, A006052, A007016, A120732, A319056, A320451, A321718, A321719, A321722, A321733, A321734, A321735, A321736, A321739.
%K nonn
%O 0,3
%A _Gus Wiseman_, Nov 18 2018
%E a(7) onwards from _Ludovic Schwob_, Apr 03 2024
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