OFFSET
1,2
REFERENCES
F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, pp. 124 and 241.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
Keith Briggs, Table of n, a(n) for n = 1..64
P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
Alexander G. Ginsberg, Firing-Rate Models in Computational Neuroscience: New Applications and Methodologies, Ph. D. Dissertation, Univ. Michigan, 2023. See p. 7.
Martin Golubitsky and Yangyang Wang, Infinitesimal homeostasis in three-node input-output networks, Journal of Mathematical Biology (2020) Vol. 80, 1163-1185.
A. Iványi, Leader election in synchronous networks, Acta Univ. Sapientiae, Mathematica, 5, 2 (2013) 54-82.
X. Li, D. S. Stones, H. Wang, H. Deng, X. Liu and G. Wang, NetMODE: Network Motif Detection without Nauty, PLoS ONE 7(12): e50093. doi:10.1371/journal.pone.0050093. - From N. J. A. Sloane, Feb 02 2013
Eric Weisstein's World of Mathematics, Weakly Connected Digraph
FORMULA
a(n) = (1/n)*Sum_{d|n} mu(n/d)*A003084(d), where mu is Moebius function.
Inverse Euler transform of A000273. - Andrew Howroyd, Dec 27 2021
MAPLE
# The function EulerInvTransform is defined in A358451.
a := EulerInvTransform(A000273):
seq(a(n), n = 1..13); # Peter Luschny, Nov 21 2022
MATHEMATICA
Needs["Combinatorica`"]; d[n_] := GraphPolynomial[n, x, Directed] /. x -> 1; max = 13; se = Series[ Sum[a[n]*x^n/n, {n, 1, max}] - Log[1 + Sum[ d[n]*x^n, {n, 1, max}]], {x, 0, max}]; sol = SolveAlways[ se == 0, x]; Do[ A003084[n] = a[n] /. sol[[1]], {n, 1, max}]; ClearAll[a, d]; a[n_] := (1/n)*Sum[ MoebiusMu[ n/d ] * A003084[d], {d, Divisors[n]} ]; Table[ a[n], {n, 1, max}] (* Jean-François Alcover, Feb 01 2012, after formula *)
terms = 13;
permcount[v_] := Module[{m = 1, s = 0, k = 0, t}, For[i = 1, i <= Length[v], i++, t = v[[i]]; k = If[i > 1 && t == v[[i - 1]], k + 1, 1]; m *= t*k; s += t]; s!/m];
edges[v_] := Sum[2*GCD[v[[i]], v[[j]]], {i, 2, Length[v]}, {j, 1, i - 1}] + Total[v - 1];
d[n_] := (s = 0; Do[s += permcount[p]*2^edges[p], {p, IntegerPartitions[n]} ]; s/n!);
A003084 = CoefficientList[Log[Sum[d[n] x^n, {n, 0, terms+1}]] + O[x]^(terms + 1), x] Range[0, terms] // Rest;
a[n_] := (1/n)*Sum[MoebiusMu[n/d] * A003084[[d]], {d, Divisors[n]}];
Table[a[n], {n, 1, terms}] (* Jean-François Alcover, Aug 30 2019, after Andrew Howroyd in A003084 *)
PROG
(Python)
from functools import lru_cache
from itertools import product, combinations
from fractions import Fraction
from math import prod, gcd, factorial
from sympy import mobius, divisors
from sympy.utilities.iterables import partitions
def A003085(n):
@lru_cache(maxsize=None)
def b(n): return int(sum(Fraction(1<<sum(p[r]*p[s]*gcd(r, s)<<1 for r, s in combinations(p.keys(), 2))+sum(r*(q*r-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 # Chai Wah Wu, Jul 05 2024
CROSSREFS
KEYWORD
nonn,nice
AUTHOR
EXTENSIONS
More terms from Vladeta Jovovic, Jan 09 2000
STATUS
approved