A178315 E.g.f.: A(x) = sqrt( Sum_{n>=0} 2^(n(n+1)/2) * x^n/n! ).

1, 1, 3, 23, 393, 13729, 943227, 126433847, 33245947857, 17276815511041, 17836691600303283, 36694285316980381463, 150671768689108469724633, 1235972596853128519493249569, 20265064539085026367759911941547, 664309630995695142408442512638430647

Offset: 0

Author

Paul D. Hanna, May 24 2010

Keywords

• nonn

Comments

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)

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)

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)

Examples

			E.g.f.: A(x) = 1 + x + 3*x^2/2! + 23*x^3/3! + 393*x^4/4! +...
A(x)^2 = 1 + 2*x + 2^3*x^2/2! + 2^6*x^3/3! + 2^10*x^4/4! +...
		

References

• F. Harary, On the notion of balance of a signed graph, Michigan Math. J. 2 (1953/54), 143-146.
• F. Harary and E. M. Palmer, On the number of balanced signed graphs, Bulletin of Mathematical Biophysics 29 (1967), 759-765.

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..81
• R. Stanley et al., The "square root" of a graph?, MathOverflow, 2021.
• F. Ardila, F. Castillo, Federico, and M. Henley, The arithmetic Tutte polynomials of the classical root systems, Int. Math. Res. Not. IMRN 12 (2015), 3830-3877.

Crossrefs

Column k=2 of A346061.

Programs

Maple

a:= n-> n!*coeff(series(add(2^binomial(j+1, 2)
        *x^j/j!, j=0..n)^(1/2), x, n+1), x, n):
seq(a(n), n=0..18);  # Alois P. Heinz, Mar 15 2021

PARI

{a(n)=n!*polcoeff(sqrt(sum(m=0,n,2^(m*(m+1)/2)*x^m/m!)+x*O(x^n)),n)}

Extensions

Edited by Max Alekseyev, Mar 13 2021