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

1,10

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.

Links

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

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!.

Example

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]

Prog

(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))}

Crossrefs

Row sums are A370059.

Column sums are A100743.

Main diagonal is A001205.

Cf. A369928, A369932 (unlabeled).

Sequence in context: A161837 A299163 A326128 * A160086 A115869 A115859

Adjacent sequences: A369928 A369929 A369930 * A369932 A369933 A369934

Keyword

nonn,tabl

Author

Andrew Howroyd, Feb 08 2024

Status

approved