A060514 Triangle T(n,k) of series-reduced (or homeomorphically irreducible) labeled graphs with n nodes and k edges, k=0..binomial(n,2).

1, 1, 1, 1, 1, 3, 0, 0, 1, 6, 3, 4, 0, 0, 1, 1, 10, 15, 20, 5, 0, 5, 20, 15, 10, 1, 1, 15, 45, 75, 90, 96, 135, 315, 510, 760, 843, 765, 395, 105, 15, 1, 1, 21, 105, 245, 525, 777, 1302, 3045, 7455, 16275, 30135, 50190, 70805, 81690, 70605, 43239, 18774, 5880, 1330

Offset: 0

Author

Vladeta Jovovic, Mar 23 2001

Keywords

• nonn
• tabf

Examples

			Triangle begins:
[1],
[1],
[1, 1],
[1, 3, 0, 0],
[1, 6, 3, 4, 0, 0, 1],
[1, 10, 15, 20, 5, 0, 5, 20, 15, 10, 1],
[1, 15, 45, 75, 90, 96, 135, 315, 510, 760, 843, 765, 395, 105, 15, 1],
[1, 21, 105, 245, 525, 777, 1302, 3045, 7455, 16275, 30135, 50190, 70805, 81690, 70605, 43239, 18774, 5880, 1330, 210, 21, 1],
...
		

References

• I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, John Wiley and Sons, N.Y., 1983.

Links

• D. M. Jackson and J. W. Reilly, The enumeration of homeomorphically irreducible labeled graphs, J. Combin. Theory, B 19 (1975), 272-286.

Crossrefs

Row sums: A003514.

For connected graphs see A331437, A331438.

Formula

E.g.f. : (1+x*y)^(-1/2)*exp(x*y/2-x^2*y^2/4)*Sum_{k=0..inf}((1+x)*exp(-x^2*y/(1+x*y)))^binomial(k, 2)*(exp(1/2*x^3*y^2/(1+x*y)))^k*x^k/k!