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A003085 Number of connected digraphs with n nodes.
(Formerly M2067)
15
1, 2, 13, 199, 9364, 1530843, 880471142, 1792473955306, 13026161682466252, 341247400399400765678, 32522568098548115377595264, 11366712907233351006127136886487, 14669074325902449468573755897547924182 (list; graph; refs; listen; history; text; internal format)
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.

Martin Golubitsky, 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.

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

Cf. A000273, A003084.

Sequence in context: A049512 A003507 A307066 * A053598 A193550 A102585

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

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Last modified November 27 23:47 EST 2020. Contains 338685 sequences. (Running on oeis4.)