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A367142
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Number of connected simple graphs on n unlabeled vertices without universal vertices.
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2
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1, 0, 0, 0, 2, 10, 78, 697, 10073, 248734, 11441903, 994695397, 163040832612, 50170816696627, 28952985431480109, 31368326987104006472, 63938133627255371867509, 245807830666379498961633640, 1787085789384745555957516856804, 24634233851674722043622102881490796
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OFFSET
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0,5
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COMMENTS
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A universal vertex is adjacent to every other vertex.
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LINKS
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FORMULA
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a(n) = Sum_{k=2..n-2} A332760(n,k) for n > 0.
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EXAMPLE
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The a(4) = 2 graphs are P_4 (path graph) and C_4 (cycle graph).
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PROG
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(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
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
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CROSSREFS
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A002494 is the not necessarily connected case.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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