A060516 -id:A060516 - cp's OEIS frontend

A060579 Number of homeomorphically irreducible general graphs on 4 labeled nodes and with n edges.

1, 6, 19, 68, 242, 704, 1981, 5140, 12364, 27614, 57598, 113108, 210812, 375606, 643646, 1066196, 1714445, 2685464, 4109493, 6158768, 9058119, 13097592, 18647371, 26175300, 36267330, 49651242, 67224024, 90083308, 119563302

Offset: 0

Vladeta Jovovic, Apr 03 2001

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

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

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

G.f.: (4*x^15 + 5*x^14 - 194*x^13 + 881*x^12 - 2058*x^11 + 3096*x^10 - 3330*x^9 + 2628*x^8 - 1398*x^7 + 359*x^6 + 72*x^5 - 93*x^4 + 28*x^3 + 4*x^2 - 4*x + 1)/(x - 1)^10. 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!.

A060580 Number of homeomorphically irreducible general graphs on 5 labeled nodes and with n edges.

Original entry on oeis.org

1, 10, 40, 185, 765, 2845, 10220, 33885, 105185, 305465, 830811, 2119875, 5091525, 11565505, 24977315, 51552005, 102175360, 195301015, 361365695, 649360880, 1136438375, 1941722170, 3245874555, 5318438260, 8555568895, 13531506921

Offset: 0

Vladeta Jovovic, Apr 03 2001

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

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

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

G.f.: - (5*x^22 - 20*x^21 + 23*x^20 - 815*x^19 + 8110*x^18 - 37255*x^17 + 104890*x^16 - 204469*x^15 + 296720*x^14 - 337455*x^13 + 310150*x^12 - 229885*x^11 + 131054*x^10 - 50485*x^9 + 6490*x^8 + 7255*x^7 - 6730*x^6 + 3242*x^5 - 995*x^4 + 180*x^3 - 5*x^2 - 5*x + 1)/(x - 1)^15. 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!.

cp's OEIS Frontend

A060579 Number of homeomorphically irreducible general graphs on 4 labeled nodes and with n edges.

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