A369931 Triangle read by rows: T(n,k) is the number of labeled simple graphs with n edges and k vertices and without endpoints or isolated vertices.

0, 0, 0, 0, 0, 1, 0, 0, 0, 3, 0, 0, 0, 6, 12, 0, 0, 0, 1, 85, 70, 0, 0, 0, 0, 100, 990, 465, 0, 0, 0, 0, 45, 2805, 11550, 3507, 0, 0, 0, 0, 10, 3595, 59990, 140420, 30016, 0, 0, 0, 0, 1, 2697, 147441, 1174670, 1802682, 286884, 0, 0, 0, 0, 0, 1335, 222516, 4710300, 22467312, 24556140, 3026655

Offset: 1

Author

Andrew Howroyd, Feb 08 2024

Keywords

• nonn
• tabl

Comments

T(n,k) is the number of traceless symmetric binary matrices with 2n 1's and k rows and at least two 1's in every row.

Examples

			Triangle begins:
  0;
  0, 0;
  0, 0, 1;
  0, 0, 0, 3;
  0, 0, 0, 6,  12;
  0, 0, 0, 1,  85,   70;
  0, 0, 0, 0, 100,  990,    465;
  0, 0, 0, 0,  45, 2805,  11550,    3507;
  0, 0, 0, 0,  10, 3595,  59990,  140420,   30016;
  0, 0, 0, 0,   1, 2697, 147441, 1174670, 1802682, 286884;
  ...
The T(3,3) = 1 matrix is:
  [0 1 1]
  [1 0 1]
  [1 1 0]
The T(4,4) = 3 matrices are:
  [0 0 1 1]  [0 1 0 1]  [0 1 1 0]
  [0 0 1 1]  [1 0 1 0]  [1 0 0 1]
  [1 1 0 0]  [0 1 0 1]  [1 0 0 1]
  [1 1 0 0]  [1 0 1 0]  [0 1 1 0]
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..1275 (first 50 rows)

Crossrefs

Row sums are A370059.

Column sums are A100743.

Main diagonal is A001205.

Cf. A369928, A369932 (unlabeled).

Programs

PARI

G(n)={my(A=x/exp(x*y + O(x*x^n))); exp(y*x^2/2 - x + O(x*x^n)) * sum(k=0, n, (1 + y + O(y*y^n))^binomial(k, 2)*A^k/k!)}
T(n)={my(r=Vec(substvec(serlaplace(G(n)), [x, y], [y, x]))); vector(#r-1, i, Vecrev(Pol(r[i+1]/y), i))}

Formula

T(n,k) = k!*[x^k][y^n] exp(y*x^2/2 - x) * Sum_{j>=0} (1 + y)^binomial(j, 2)*(x/exp(y*x))^j/j!.