A060577 - OEIS

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

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OFFSET

0,3

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..52.

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).

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)

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 *)

CROSSREFS

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

Sequence in context: A343345 A343346 A296468 * A360042 A360361 A197985

Adjacent sequences: A060574 A060575 A060576 * A060578 A060579 A060580