A084546 Triangle read by rows: T(n,k) = C( C(n,2), k) for n >= 0, 0 <= k <= C(n,2).

1, 1, 1, 1, 1, 3, 3, 1, 1, 6, 15, 20, 15, 6, 1, 1, 10, 45, 120, 210, 252, 210, 120, 45, 10, 1, 1, 15, 105, 455, 1365, 3003, 5005, 6435, 6435, 5005, 3003, 1365, 455, 105, 15, 1, 1, 21, 210, 1330, 5985, 20349, 54264, 116280, 203490, 293930, 352716, 352716, 293930, 203490, 116280, 54264, 20349, 5985, 1330, 210, 21, 1

Offset

0,6

Comments

T(n,k) gives number of labeled simple graphs with n nodes and k edges.

References

J. L. Gross and J. Yellen, eds., Handbook of Graph Theory, CRC Press, 2004; p. 517.

Links

Alois P. Heinz, Rows n = 0..42, flattened

R. J. Mathar, Statistics on Small Graphs, arXiv:1709.09000 (2017) table 66.

Example

Triangle begins:
  1;
  1;
  1, 1;
  1, 3,  3,  1;
  1, 6, 15, 20, 15, 6, 1;
  ...

Maple

C:= binomial:
T:= (n, k)-> C( C(n, 2), k):
seq(seq(T(n, k), k=0..C(n, 2)), n=0..10);  # Alois P. Heinz, Feb 17 2023

Mathematica

Table[Table[Binomial[Binomial[n, 2], k], {k, 0, Binomial[n, 2]}], {n, 1, 7}]//Grid (* Geoffrey Critzer, Apr 28 2011 *)

Crossrefs

Cf. A083029. A subset of the rows of Pascal's triangle A007318.

Cf. A006125 (row sums), A008406 (unlabeled graphs).

Main diagonal gives A116508.

Sequence in context: A080858 A144228 A083029 * A288266 A174116 A270273

Adjacent sequences: A084543 A084544 A084545 * A084547 A084548 A084549

Keyword

nonn,tabf

Author

N. J. A. Sloane, Jul 13 2003

Extensions

T(0,0)=1 prepended by Alois P. Heinz, Feb 17 2023

Status

approved