OFFSET
0,5
COMMENTS
A universal vertex is adjacent to every other vertex.
LINKS
Chai Wah Wu, Table of n, a(n) for n = 0..87
Wikipedia, Vertex_(graph_theory).
EXAMPLE
The a(4) = 2 graphs are P_4 (path graph) and C_4 (cycle graph).
PROG
(Python)
from functools import lru_cache
from itertools import combinations
from fractions import Fraction
from math import prod, gcd, factorial
from sympy import mobius, divisors
from sympy.utilities.iterables import partitions
def A367142(n):
if n == 0: return 1
@lru_cache(maxsize=None)
def b(n): return int(sum(Fraction(1<<sum(p[r]*p[s]*gcd(r, s) for r, s in combinations(p.keys(), 2))+sum((q>>1)*r+(q*r*(r-1)>>1) for q, r in p.items()), prod(q**r*factorial(r) for q, r in p.items())) for p in partitions(n)))
@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-b(n-1) # Chai Wah Wu, Jul 03 2024
CROSSREFS
KEYWORD
nonn
AUTHOR
Andrew Howroyd, Nov 06 2023
STATUS
approved