

A053419


Number of graphs with loops (symmetric relations) with n edges.


12



1, 2, 5, 14, 38, 107, 318, 972, 3111, 10410, 36371, 132656, 504636, 1998361, 8224448, 35112342, 155211522, 709123787, 3342875421, 16234342515, 81102926848, 416244824068, 2192018373522, 11831511359378, 65387590986455, 369661585869273, 2135966349269550, 12604385044890628
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

0,2


COMMENTS

In a multiset partition, two vertices are equivalent if in every block the multiplicity of the first is equal to the multiplicity of the second. a(n) is the number of nonisomorphic multiset partitions of {1, 1, 2, 2, 3, 3, ..., n, n} with no equivalent vertices. For example, nonisomorphic representatives of the a(2) = 5 multiset partitions are (1)(122), (11)(22), (1)(1)(22), (1)(2)(12), (1)(1)(2)(2).  Gus Wiseman, Jul 18 2018
a(n) is the number of unlabeled simple graphs with n edges rooted at one vertex.  Andrew Howroyd, Nov 22 2020


LINKS

Andrew Howroyd, Table of n, a(n) for n = 0..50


FORMULA

Euler transform of A191970.  Andrew Howroyd, Oct 22 2019


MATHEMATICA

seq[n_] := Module[{A = O[x]^n}, G[2n, x+A, {1}] // CoefficientList[#, x]&]; (* JeanFrançois Alcover, Dec 03 2020, using Andrew Howroyd's code for g.f. G in A339063 *)


PROG

(PARI) \\ See A339063 for G.
seq(n)={my(A=O(x*x^n)); Vec(G(2*n, x+A, [1]))} \\ Andrew Howroyd, Nov 22 2020


CROSSREFS

Cf. A000664, A000666, A007716, A007717, A020555, A050535, A053419, A094574, A191970 (multisets), A316974, A339063.
Sequence in context: A026288 A047086 A006574 * A079227 A148314 A001011
Adjacent sequences: A053416 A053417 A053418 * A053420 A053421 A053422


KEYWORD

nonn


AUTHOR

Vladeta Jovovic, Jan 10 2000


EXTENSIONS

a(16)a(24) from Max Alekseyev, Jan 22 2010
Terms a(25) and beyond from Andrew Howroyd, Oct 22 2019


STATUS

approved



