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

0,2

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

Table of n, a(n) for n=0..36.

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

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!.

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 *)

Crossrefs

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

Sequence in context: A007056 A026551 A296719 * A147078 A341704 A146416

Adjacent sequences: A060575 A060576 A060577 * A060579 A060580 A060581

Keyword

easy,nonn

Author

Vladeta Jovovic, Apr 03 2001

Status

approved