A003085 Number of weakly connected digraphs with n unlabeled nodes. (Formerly M2067)

1, 2, 13, 199, 9364, 1530843, 880471142, 1792473955306, 13026161682466252, 341247400399400765678, 32522568098548115377595264, 11366712907233351006127136886487, 14669074325902449468573755897547924182

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 *)

Crossrefs

Row sums of A054733.

Column sums of A350789.

The labeled case is A003027.

Cf. A000273, A003084, A035512 (strongly connected).

Sequence in context: A338979 A307066 A361368 * A053598 A193550 A367830

Adjacent sequences: A003082 A003083 A003084 * A003086 A003087 A003088

Keyword

nonn,nice

Author

N. J. A. Sloane

Extensions

More terms from Vladeta Jovovic, Jan 09 2000

Status

approved