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A334335 Inverse Euler transform of A000568.

%I A334335 #34 Jul 04 2024 02:23:16
%S A334335 1,0,1,2,8,43,398,6413,184596,9540998,894012628,153204356304,
%T A334335 48387996396590,28352880689501075,30992600581556641380,
%U A334335 63499394791445838416399,244849247994227524521679624,1783153933475289754036309451798,24603857772350530383609071261316942
%N A334335 Inverse Euler transform of A000568.
%C A334335 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]
%H A334335 Pontus von Brömssen, <a href="/A334335/b334335.txt">Table of n, a(n) for n = 1..76</a>
%H A334335 Pontus Andersson (von Brömssen), <a href="https://doi.org/10.1002/(SICI)1098-2418(199810/12)13:3/4%3C249::AID-RSA4%3E3.0.CO;2-V">On the asymptotic distributions of subgraph counts in a random tournament</a>, Random Structures & Algorithms 13 (1998), 249-260.
%H A334335 Gordon F. Royle, Cheryl E. Praeger, S. P. Glasby, Saul D. Freedman, and Alice Devillers, <a href="https://doi.org/10.1007/s10801-022-01197-0">Tournaments and even graphs are equinumerous</a>, Journal of Algebraic Combinatorics 57 (2023), 515-524; <a href="https://arxiv.org/abs/2204.01947">arXiv version</a>, arXiv:2204.01947 [math.CO], 2022.
%o A334335 (Python)
%o A334335 from functools import lru_cache
%o A334335 from itertools import product
%o A334335 from fractions import Fraction
%o A334335 from math import prod, gcd, factorial
%o A334335 from sympy import mobius, divisors
%o A334335 from sympy.utilities.iterables import partitions
%o A334335 def A334335(n):
%o A334335     @lru_cache(maxsize=None)
%o A334335     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)))
%o A334335     @lru_cache(maxsize=None)
%o A334335     def c(n): return n*b(n)-sum(c(k)*b(n-k) for k in range(1,n))
%o A334335     return sum(mobius(n//d)*c(d) for d in divisors(n,generator=True))//n # _Chai Wah Wu_, Jul 03 2024
%Y A334335 Cf. A000568.
%K A334335 nonn
%O A334335 1,4
%A A334335 _Pontus von Brömssen_, Apr 23 2020