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

0,3

Comments

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

References

Ian P. Goulden and David M. Jackson, Combinatorial Enumeration, Wiley, N.Y., 1983.

Links

Table of n, a(n) for n=0..52.

Vladeta Jovovic, Generating functions for homeomorphically irreducible general graphs on n labeled nodes.

Vladeta 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)

Sum_{n>=0} 1/a(n) = 10/3 + (2*Pi/sqrt(33))*tan(Pi*sqrt(33)/2). - Amiram Eldar, Dec 06 2025

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

Keyword

nonn,easy

Author

Vladeta Jovovic, Apr 04 2001

Extensions

More terms from James Sellers, Apr 04 2001

Status

approved