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A060577 Number of homeomorphically irreducible general graphs on 2 labeled nodes and with n edges. 1
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 (list; graph; refs; listen; history; text; internal format)
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: A190564 A008369 A296468 * A197985 A058579 A022318

Adjacent sequences:  A060574 A060575 A060576 * A060578 A060579 A060580

KEYWORD

nonn,easy

AUTHOR

Vladeta Jovovic, Apr 04 2001

EXTENSIONS

More terms from James A. Sellers, Apr 04 2001

STATUS

approved

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Last modified January 22 16:37 EST 2020. Contains 331152 sequences. (Running on oeis4.)