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A135588
Number of symmetric (0,1)-matrices with exactly n entries equal to 1 and no zero rows or columns.
11
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
0,3
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..721 (first 51 terms from Vincenzo Librandi)
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)).
EXAMPLE
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)
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 *)
KEYWORD
nonn
AUTHOR
Vladeta Jovovic, Feb 25 2008, Mar 03 2008, Mar 04 2008
STATUS
approved