A276639 Triangle T(m, n) = the number of point-labeled graphs with n points and m edges, no points isolated. By rows, n >= 0, ceiling(n/2) <= m <= binomial(n,2).

1, 1, 3, 1, 3, 16, 15, 6, 1, 30, 135, 222, 205, 120, 45, 10, 1, 15, 330, 1581, 3760, 5715, 6165, 4945, 2997, 1365, 455, 105, 15, 1, 315, 4410, 23604, 73755, 159390, 259105, 331716, 343161, 290745, 202755, 116175, 54257, 20349, 5985, 1330, 210, 21, 1

Offset: 1

Author

David Pasino, Sep 08 2016

Keywords

• nonn
• tabf

Comments

The row sums are A006129, omitting row 1 and A006129(1).

Examples

			Triangle T(n, m) begins:
n/m  0  1  2  3   4    5    6    7    8    9   10
0    1  0  0  0   0    0    0    0    0    0   0
1    0  0  0  0   0    0    0    0    0    0   0
2    0  1  0  0   0    0    0    0    0    0   0
3    0  0  3  1   0    0    0    0    0    0   0
4    0  0  3  16  15   6    1    0    0    0   0
5    0  0  0  30  135  222  205  120  45   10  1
		

Links

• David Pasino, Table of n, a(n) for n = 1..512

Crossrefs

Another version is A054548.

Programs

Mathematica

Table[Sum[Binomial[n, k] (-1)^(n - k) Binomial[Binomial[k, 2], m], {k, 0, n}], {n, 7}, {m, Ceiling[n/2], Binomial[n, 2]}] /. {} -> {1} // Flatten (* Michael De Vlieger, Sep 19 2016 *)

Formula

T(n, m) = Sum_{k=0,..n} binomial(n, k) * (-1)^(n-k) * A084546(k, m).