cp's OEIS Frontend

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

%I A135588 #17 Jan 22 2024 17:45:38
%S A135588 1,1,2,6,20,74,302,1314,6122,29982,154718,831986,4667070,27118610,
%T A135588 163264862,1013640242,6488705638,42687497378,288492113950,
%U A135588 1998190669298,14177192483742,102856494496050,762657487965086,5771613810502002,44555989658479726,350503696871063138
%N A135588 Number of symmetric (0,1)-matrices with exactly n entries equal to 1 and no zero rows or columns.
%H A135588 Alois P. Heinz, <a href="/A135588/b135588.txt">Table of n, a(n) for n = 0..721</a> (first 51 terms from Vincenzo Librandi)
%F A135588 G.f.: Sum_{n>=0} (1+x)^n*(1+x^2)^binomial(n,2)/2^(n+1).
%F A135588 G.f.: Sum_{n>=0} (Sum_{k=0..n} (-1)^(n-k)*binomial(n,k)*(1+x)^k*(1+x^2)^binomial(k,2)).
%e A135588 From _Gus Wiseman_, Nov 14 2018: (Start)
%e A135588 The a(4) = 20 matrices:
%e A135588   [11]
%e A135588   [11]
%e A135588 .
%e A135588   [110][101][100][100][011][010][010][001][001]
%e A135588   [100][010][011][001][100][110][101][010][001]
%e A135588   [001][100][010][011][100][001][010][101][110]
%e A135588 .
%e A135588   [1000][1000][1000][1000][0100][0100][0010][0010][0001][0001]
%e A135588   [0100][0100][0010][0001][1000][1000][0100][0001][0100][0010]
%e A135588   [0010][0001][0100][0010][0010][0001][1000][1000][0010][0100]
%e A135588   [0001][0010][0001][0100][0001][0010][0001][0100][1000][1000]
%e A135588 (End)
%t A135588 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 *)
%t A135588 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 *)
%Y A135588 Row sums of A135589.
%Y A135588 Cf. A049311, A054976, A101370, A104601, A104602, A120733, A138178, A283877, A316983, A320796, A321401, A321405.
%K A135588 nonn
%O A135588 0,3
%A A135588 _Vladeta Jovovic_, Feb 25 2008, Mar 03 2008, Mar 04 2008