OFFSET
1,4
COMMENTS
It appears that a(n) is the number of connected even graphs with n vertices, as defined in the Andersson paper (verified for n <= 10): a graph G is even if, for a given orientation of its edges, all automorphisms of G reverse the orientation of an even number of edges. If this is true, it means that A000568(n) is the number of even graphs (not necessarily connected) with n vertices. [This has been proved by Royle et al. 2023. - Pontus von Brömssen, Apr 06 2022]
LINKS
Pontus von Brömssen, Table of n, a(n) for n = 1..76
Pontus Andersson (von Brömssen), On the asymptotic distributions of subgraph counts in a random tournament, Random Structures & Algorithms 13 (1998), 249-260.
Gordon F. Royle, Cheryl E. Praeger, S. P. Glasby, Saul D. Freedman, and Alice Devillers, Tournaments and even graphs are equinumerous, Journal of Algebraic Combinatorics 57 (2023), 515-524; arXiv version, arXiv:2204.01947 [math.CO], 2022.
PROG
(Python)
from functools import lru_cache
from itertools import product
from fractions import Fraction
from math import prod, gcd, factorial
from sympy import mobius, divisors
from sympy.utilities.iterables import partitions
def A334335(n):
@lru_cache(maxsize=None)
def b(n): return int(sum(Fraction(1<<(sum(p[r]*p[s]*gcd(r, s) for r, s in product(p.keys(), repeat=2))-sum(p.values())>>1), prod(q**p[q]*factorial(p[q]) for q in p)) for p in partitions(n) if all(q&1 for q in p)))
@lru_cache(maxsize=None)
def c(n): return n*b(n)-sum(c(k)*b(n-k) for k in range(1, n))
return sum(mobius(n//d)*c(d) for d in divisors(n, generator=True))//n # Chai Wah Wu, Jul 03 2024
CROSSREFS
KEYWORD
nonn
AUTHOR
Pontus von Brömssen, Apr 23 2020
STATUS
approved