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%I #23 Dec 03 2020 07:49:50
%S 1,2,5,14,38,107,318,972,3111,10410,36371,132656,504636,1998361,
%T 8224448,35112342,155211522,709123787,3342875421,16234342515,
%U 81102926848,416244824068,2192018373522,11831511359378,65387590986455,369661585869273,2135966349269550,12604385044890628
%N Number of graphs with loops (symmetric relations) with n edges.
%C 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 non-isomorphic multiset partitions of {1, 1, 2, 2, 3, 3, ..., n, n} with no equivalent vertices. For example, non-isomorphic 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
%C a(n) is the number of unlabeled simple graphs with n edges rooted at one vertex. - _Andrew Howroyd_, Nov 22 2020
%H Andrew Howroyd, <a href="/A053419/b053419.txt">Table of n, a(n) for n = 0..50</a>
%F Euler transform of A191970. - _Andrew Howroyd_, Oct 22 2019
%t seq[n_] := Module[{A = O[x]^n}, G[2n, x+A, {1}] // CoefficientList[#, x]&]; (* _Jean-François Alcover_, Dec 03 2020, using _Andrew Howroyd_'s code for g.f. G in A339063 *)
%o (PARI) \\ See A339063 for G.
%o seq(n)={my(A=O(x*x^n)); Vec(G(2*n, x+A, [1]))} \\ _Andrew Howroyd_, Nov 22 2020
%Y Cf. A000664, A000666, A007716, A007717, A020555, A050535, A053419, A094574, A191970 (multisets), A316974, A339063.
%K nonn
%O 0,2
%A _Vladeta Jovovic_, Jan 10 2000
%E a(16)-a(24) from _Max Alekseyev_, Jan 22 2010
%E Terms a(25) and beyond from _Andrew Howroyd_, Oct 22 2019
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