A060577 Number of homeomorphically irreducible general graphs on 2 labeled nodes and with n edges.

1, 1, 4, 6, 11, 17, 24, 32, 41, 51, 62, 74, 87, 101, 116, 132, 149, 167, 186, 206, 227, 249, 272, 296, 321, 347, 374, 402, 431, 461, 492, 524, 557, 591, 626, 662, 699, 737, 776, 816, 857, 899, 942, 986, 1031, 1077, 1124, 1172, 1221, 1271, 1322, 1374, 1427

Offset: 0

Author

Vladeta Jovovic, Apr 04 2001

Keywords

• nonn
• easy

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
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Cf. A003514, A046691, A060516, A060533-A060537, A060576-A060581.

Programs

Maple

gf := (2*x^5 - 4*x^4 + 4*x^3 - 4*x^2 + 2*x - 1)/(x - 1)^3: s := series(gf, x, 100): for i from 0 to 100 do printf(`%d,`,coeff(s,x,i)) od:

Mathematica

Join[{1, 1, 4}, Table[n (n + 3)/2 - 3, {n, 3, 60}]] (* Bruno Berselli, Aug 20 2015 *)

Formula

G.f.: (2*x^5 - 4*x^4 + 4*x^3 - 4*x^2 + 2*x - 1)/(x - 1)^3.

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!.

From Marco Ripà, Aug 20 2015: (Start)

a(n) = ceiling( (1/2)*(3*n^2 - 10*n + 9)/(n - 2) ) + floor( (3/2)*(n-1)^2 ) - n^2 + 3*n - 3 with n > 2, a(0) = a(1) = 1, a(2) = 4.

a(n) = n*(n + 3)/2 - 3 for n > 2.

a(n) = A046691(n-1) for n > 2. (End)

Extensions

More terms from James Sellers, Apr 04 2001