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%I #14 Jul 03 2021 09:03:08
%S 1,1,3,23,393,13729,943227,126433847,33245947857,17276815511041,
%T 17836691600303283,36694285316980381463,150671768689108469724633,
%U 1235972596853128519493249569,20265064539085026367759911941547,664309630995695142408442512638430647
%N E.g.f.: A(x) = sqrt( Sum_{n>=0} 2^(n(n+1)/2) * x^n/n! ).
%C Number of equivalence classes of graphs on n vertices, where two graphs are in the same class if one can be obtained from the other by loop-switching a subset of its connected components. Here, loop-switching is a fixed-point-free involution adding a loop to every vertex that doesn't have one while simultaneously deleting the loops from all vertices that do. (see MO link)
%C Also, number of balanced signed graphs (without loops) on n vertices. A graph is signed if every edge has a sign, either positive or negative, and it is balanced if every cycle has an even number of negative edges. (see MO link)
%C Also, number of graphs on vertices {1,2,...,n} with loops allowed, where the least vertex in each component has a loop. (see MO link)
%D F. Harary, On the notion of balance of a signed graph, Michigan Math. J. 2 (1953/54), 143-146.
%D F. Harary and E. M. Palmer, On the number of balanced signed graphs, Bulletin of Mathematical Biophysics 29 (1967), 759-765.
%H Alois P. Heinz, <a href="/A178315/b178315.txt">Table of n, a(n) for n = 0..81</a>
%H R. Stanley et al., <a href="https://mathoverflow.net/q/385303">The "square root" of a graph?</a>, MathOverflow, 2021.
%H F. Ardila, F. Castillo, Federico, and M. Henley, <a href="https://arxiv.org/abs/1305.6621">The arithmetic Tutte polynomials of the classical root systems</a>, Int. Math. Res. Not. IMRN 12 (2015), 3830-3877.
%e E.g.f.: A(x) = 1 + x + 3*x^2/2! + 23*x^3/3! + 393*x^4/4! +...
%e A(x)^2 = 1 + 2*x + 2^3*x^2/2! + 2^6*x^3/3! + 2^10*x^4/4! +...
%p a:= n-> n!*coeff(series(add(2^binomial(j+1, 2)
%p *x^j/j!, j=0..n)^(1/2), x, n+1), x, n):
%p seq(a(n), n=0..18); # _Alois P. Heinz_, Mar 15 2021
%o (PARI) {a(n)=n!*polcoeff(sqrt(sum(m=0,n,2^(m*(m+1)/2)*x^m/m!)+x*O(x^n)),n)}
%Y Column k=2 of A346061.
%K nonn
%O 0,3
%A _Paul D. Hanna_, May 24 2010
%E Edited by _Max Alekseyev_, Mar 13 2021
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