This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A370723 #43 May 20 2024 10:30:10
%S A370723 1,2,5,14,39,123,393,1352,4782,17824,68481,274166
%N A370723 Number of symmetric (0,1)-matrices with sum of entries equal to n and no zero rows or columns, with weakly decreasing row sums and column sums.
%e A370723 The a(3) = 5 matrices:
%e A370723 [1 0 0] [1 0 0] [0 1 0] [0 0 1] [1 1]
%e A370723 [0 1 0] [0 0 1] [1 0 0] [0 1 0] [1 0]
%e A370723 [0 0 1] [0 1 0] [0 0 1] [1 0 0]
%t A370723 a[0] = 1;
%t A370723 a[n_] := a[n] = Length[Select[Subsets[Tuples[Range[n], 2], {n}], Module[{matrix, rows, cols}, matrix = ConstantArray[0, {n, n}]; (matrix[[#[[1]], #[[2]]]] = 1) & /@ #; rows = Total[matrix, {2}]; cols = Total[matrix, {1}]; And[Union[First /@ #] == Range[Max @@ First /@ #], Union[Last /@ #] == Range[Max @@ Last /@ #], Sort[Reverse /@ #] == #, OrderedQ[Reverse[rows]], OrderedQ[Reverse[cols]]]] &]];
%t A370723 Table[a[n], {n, 1, 6}] (* _Robert P. P. McKone_, May 19 2024, from _Gus Wiseman_ in A135588 *)
%Y A370723 Cf. A178718, A135588, A321652, A365961.
%K A370723 nonn,more
%O A370723 1,2
%A A370723 _Ludovic Schwob_, May 18 2024