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

1,1

Links

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.

Formula

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

Mathematica

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 *)

Prog

(Python)
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<<sum(p[r]*p[s]*gcd(r, s) for r, s in combinations(p.keys(), 2))+sum(((q>>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

Crossrefs

Cf. A000666, A054291.

Sequence in context: A213704 A278099 A000171 * A289651 A342588 A302751

Adjacent sequences: A054919 A054920 A054921 * A054923 A054924 A054925

Keyword

nonn,easy,changed

Author

N. J. A. Sloane, May 24 2000

Extensions

More terms from Vladeta Jovovic, Jul 17 2000

Status

approved