A369928 Triangle read by rows: T(n,k) is the number of simple graphs on n labeled vertices with k edges and without endpoints, n >= 0, 0 <= k <= n*(n-1)/2.

1, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 4, 3, 6, 1, 1, 0, 0, 10, 15, 42, 90, 100, 45, 10, 1, 1, 0, 0, 20, 45, 162, 595, 1590, 3075, 3655, 2703, 1335, 455, 105, 15, 1, 1, 0, 0, 35, 105, 462, 2310, 9495, 32130, 85365, 166341, 231861, 237125, 184380, 111870, 53634, 20307, 5985, 1330, 210, 21, 1

Offset: 0

Author

Andrew Howroyd, Feb 07 2024

Keywords

• nonn
• tabf

Examples

			Triangle begins:
[0] 1;
[1] 1;
[2] 1, 0;
[3] 1, 0, 0,  1;
[4] 1, 0, 0,  4,  3,  6,    1;
[5] 1, 0, 0, 10, 15,  42,  90,  100,   45,   10,    1;
[6] 1, 0, 0, 20, 45, 162, 595, 1590, 3075, 3655, 2703, 1335, 455, 105, 15, 1;
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..1350 (rows 0..20)

Crossrefs

Row sums are A059167.

Cf. A084546, A123551 (unlabeled), A245796 (with endpoints).

Programs

PARI

\\ row(n) gives n-th row as vector.
row(n)={my(A=x/exp(x*y + O(x*x^n))); Vecrev(polcoef(serlaplace(exp(y*x^2/2 + O(x*x^n)) * sum(k=0, n, (1 + y)^binomial(k, 2)*A^k/k!)), n), 1 + binomial(n,2))}
{ for(n=0, 6, print(row(n))) }

Formula

T(n,k) = A084546(n,k) - A245796(n,k).

E.g.f.: exp(y*x^2/2) * Sum_{k>=0} (1 + y)^binomial(k, 2)*(x/exp(y*x))^k/k!.