login
Triangle read by rows: T(n,k) = C( C(n,2), k) for n >= 0, 0 <= k <= C(n,2).
22

%I #23 Feb 17 2023 09:10:51

%S 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,

%T 1,15,105,455,1365,3003,5005,6435,6435,5005,3003,1365,455,105,15,1,1,

%U 21,210,1330,5985,20349,54264,116280,203490,293930,352716,352716,293930,203490,116280,54264,20349,5985,1330,210,21,1

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

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

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

%H Alois P. Heinz, <a href="/A084546/b084546.txt">Rows n = 0..42, flattened</a>

%H R. J. Mathar, <a href="http://arxiv.org/abs/1709.09000">Statistics on Small Graphs</a>, arXiv:1709.09000 (2017) table 66.

%e Triangle begins:

%e 1;

%e 1;

%e 1, 1;

%e 1, 3, 3, 1;

%e 1, 6, 15, 20, 15, 6, 1;

%e ...

%p C:= binomial:

%p T:= (n, k)-> C( C(n, 2), k):

%p seq(seq(T(n, k), k=0..C(n, 2)), n=0..10); # _Alois P. Heinz_, Feb 17 2023

%t Table[Table[Binomial[Binomial[n,2],k],{k,0,Binomial[n,2]}],{n,1,7}]//Grid (* _Geoffrey Critzer_, Apr 28 2011 *)

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

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

%K nonn,tabf

%O 0,6

%A _N. J. A. Sloane_, Jul 13 2003

%E T(0,0)=1 prepended by _Alois P. Heinz_, Feb 17 2023