A135588 Number of symmetric (0,1)-matrices with exactly n entries equal to 1 and no zero rows or columns.

1, 1, 2, 6, 20, 74, 302, 1314, 6122, 29982, 154718, 831986, 4667070, 27118610, 163264862, 1013640242, 6488705638, 42687497378, 288492113950, 1998190669298, 14177192483742, 102856494496050, 762657487965086, 5771613810502002, 44555989658479726, 350503696871063138

Offset: 0

Author

Vladeta Jovovic, Feb 25 2008, Mar 03 2008, Mar 04 2008

Keywords

• nonn

Examples

			From _Gus Wiseman_, Nov 14 2018: (Start)
The a(4) = 20 matrices:
  [11]
  [11]
.
  [110][101][100][100][011][010][010][001][001]
  [100][010][011][001][100][110][101][010][001]
  [001][100][010][011][100][001][010][101][110]
.
  [1000][1000][1000][1000][0100][0100][0010][0010][0001][0001]
  [0100][0100][0010][0001][1000][1000][0100][0001][0100][0010]
  [0010][0001][0100][0010][0010][0001][1000][1000][0010][0100]
  [0001][0010][0001][0100][0001][0010][0001][0100][1000][1000]
(End)
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..721 (first 51 terms from Vincenzo Librandi)

Crossrefs

Row sums of A135589.

Cf. A049311, A054976, A101370, A104601, A104602, A120733, A138178, A283877, A316983, A320796, A321401, A321405.

Programs

Mathematica

Table[Sum[SeriesCoefficient[(1+x)^k*(1+x^2)^(k*(k-1)/2)/2^(k+1),{x,0,n}],{k,0,Infinity}],{n,0,20}] (* Vaclav Kotesovec, Jul 02 2014 *)
Join[{1},  Table[Length[Select[Subsets[Tuples[Range[n], 2], {n}], And[Union[First/@#]==Range[Max@@First/@#], Union[Last/@#]==Range[Max@@Last/@#], Sort[Reverse/@#]==#]&]], {n, 5}]] (* Gus Wiseman, Nov 14 2018 *)

Formula

G.f.: Sum_{n>=0} (1+x)^n*(1+x^2)^binomial(n,2)/2^(n+1).

G.f.: Sum_{n>=0} (Sum_{k=0..n} (-1)^(n-k)*binomial(n,k)*(1+x)^k*(1+x^2)^binomial(k,2)).