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A367142
Number of connected simple graphs on n unlabeled vertices without universal vertices.
2
1, 0, 0, 0, 2, 10, 78, 697, 10073, 248734, 11441903, 994695397, 163040832612, 50170816696627, 28952985431480109, 31368326987104006472, 63938133627255371867509, 245807830666379498961633640, 1787085789384745555957516856804, 24634233851674722043622102881490796
OFFSET
0,5
COMMENTS
A universal vertex is adjacent to every other vertex.
LINKS
FORMULA
a(n) = A001349(n) - A000088(n-1) for n > 0.
a(n) = Sum_{k=2..n-2} A332760(n,k) for n > 0.
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
A002494 is the not necessarily connected case.
Sequence in context: A134980 A355471 A240599 * A212381 A098692 A300994
KEYWORD
nonn
AUTHOR
Andrew Howroyd, Nov 06 2023
STATUS
approved