A060537 Number of homeomorphically irreducible multigraphs (or series-reduced multigraphs or multigraphs without nodes of degree 2) on 7 labeled nodes.

1, 21, 105, 266, 1386, 6678, 25403, 100506, 384678, 1393903, 4831890, 15955485, 50080478, 149211930, 421819950, 1132236630, 2890927935, 7040892159, 16411041500, 36733789575, 79230165105, 165194651065, 333926559540

Offset: 0

Author

Vladeta Jovovic, Apr 01 2001

Keywords

• easy
• nonn

References

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

Links

• Vladeta Jovovic, Generating functions for homeomorphically irreducible multigraphs on n labeled nodes

Crossrefs

Cf. A003514, A060516, A060533-A060537.

Formula

G.f.: (7*x^33 - 42*x^32 + 105*x^31 + 3598*x^30 - 64995*x^29 + 498369*x^28 - 2213029*x^27 + 6169800*x^26 - 10213560*x^25 + 4476990*x^24 + 27664014*x^23 - 97812519*x^22 + 197723150*x^21 - 296237340*x^20 + 352014180*x^19 - 334492361*x^18 + 243984426*x^17 - 117769575*x^16 + 9628325*x^15 + 45726945*x^14 - 50729175*x^13 + 31353175*x^12 - 11717370*x^11 + 1358280*x^10 + 1395765*x^9 - 1068648*x^8 + 395328*x^7 - 77805*x^6 + 882*x^5 + 4095*x^4 - 1141*x^3 + 126*x^2 - 1)/(x - 1)^21. E.g.f. for homeomorphically irreducible multigraphs 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, 2)*exp( - x^2*y*k^2/(2*(1 + x*y)) - x^2*y*k/2)*y^k/k!.