A054291 -id:A054291 - cp's OEIS frontend

A054922 Number of connected unlabeled symmetric relations (graphs with loops) having n nodes such that complement is also connected.

2, 0, 0, 10, 164, 2670, 56724, 1867860, 104538928, 10461483366, 1912179618740, 644464839239880, 402785011941549964, 468944407349226545614, 1021179521951204217530900, 4174755063830188009750183026

Offset: 1

N. J. A. Sloane, May 24 2000

Andrew Howroyd, Table of n, a(n) for n = 1..50
V. A. Liskovets, Some easily derivable sequences, J. Integer Sequences, 3 (2000), #00.2.2.

Cf. A000666, A054291.

A000666 = Cases[Import["https://oeis.org/A000666/b000666.txt", "Table"], {, }][[All, 2]];
A054921 = Cases[Import["https://oeis.org/A054921/b054921.txt", "Table"], {, }][[All, 2]];
a[n_] := 2*A054921[[n + 1]] - A000666[[n + 1]];
Array[a, 50] (* Jean-François Alcover, Aug 31 2019 *)

from functools import lru_cache
from itertools import combinations
from math import prod, factorial, gcd
from fractions import Fraction
from sympy.utilities.iterables import partitions
from sympy import mobius, divisors
def A054922(n):
    @lru_cache(maxsize=None)
    def b(n): return int(sum(Fraction(1<>1)+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(d)*c(n//d) for d in divisors(n,generator=True))//n<<1)-b(n) # Chai Wah Wu, Jul 10 2024

a(n) = 2*A054921(n) - A000666(n).

More terms from Vladeta Jovovic, Jul 17 2000

cp's OEIS Frontend

A054922 Number of connected unlabeled symmetric relations (graphs with loops) having n nodes such that complement is also connected.

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