A060578 Number of homeomorphically irreducible general graphs on 3 labeled node and with n edges.

1, 3, 9, 21, 60, 135, 282, 537, 945, 1561, 2451, 3693, 5378, 7611, 10512, 14217, 18879, 24669, 31777, 40413, 50808, 63215, 77910, 95193, 115389, 138849, 165951, 197101, 232734, 273315, 319340, 371337, 429867, 495525, 568941, 650781, 741748

Offset: 0

Author

Vladeta Jovovic, Apr 03 2001

Keywords

• easy
• nonn

Comments

A homeomorphically irreducible general graph is a graph with multiple edges and loops and without nodes of degree 2.

References

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

Links

• V. Jovovic, Generating functions for homeomorphically irreducible general graphs on n labeled nodes
• V. Jovovic, Recurrences for the numbers of homeomorphically irreducible general graphs on m labeled nodes and n edges

Crossrefs

Cf. A003514, A060516, A060533-A060537, A060576-A060581.

Programs

Mathematica

CoefficientList[Series[-(8x^9-36x^8+66x^7-70x^6+51x^5-24x^4+8x^3-6x^2+3x-1)/(x-1)^6,{x,0,40}],x] (* Harvey P. Dale, Jul 22 2018 *)

Formula

G.f.: - (8*x^9 - 36*x^8 + 66*x^7 - 70*x^6 + 51*x^5 - 24*x^4 + 8*x^3 - 6*x^2 + 3*x - 1)/(x - 1)^6. E.g.f. for homeomorphically irreducible general graphs with n nodes and k edges is (1 + x*y)^( - 1/2)*exp( - x*y/2 + x^2*y^2/4)*Sum_{k >= 0} 1/(1 - x)^binomial(k + 1, 2)*exp( - x^2*y*k^2/(2*(1 + x*y)) - x^2*y*k/2)*y^k/k!.