

A178315


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


4



1, 1, 3, 23, 393, 13729, 943227, 126433847, 33245947857, 17276815511041, 17836691600303283, 36694285316980381463, 150671768689108469724633, 1235972596853128519493249569, 20265064539085026367759911941547, 664309630995695142408442512638430647
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

0,3


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 loopswitching a subset of its connected components. Here, loopswitching is a fixedpointfree 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)


REFERENCES

F. Harary, On the notion of balance of a signed graph, Michigan Math. J. 2 (1953/54), 143146.
F. Harary and E. M. Palmer, On the number of balanced signed graphs, Bulletin of Mathematical Biophysics 29 (1967), 759765.


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), 38303877.


EXAMPLE

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! +...


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


PROG

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


CROSSREFS

Column k=2 of A346061.
Sequence in context: A243195 A233218 A114601 * A210910 A280654 A118195
Adjacent sequences: A178312 A178313 A178314 * A178316 A178317 A178318


KEYWORD

nonn


AUTHOR

Paul D. Hanna, May 24 2010


EXTENSIONS

Edited by Max Alekseyev, Mar 13 2021


STATUS

approved



