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A003027 Number of weakly connected digraphs with n labeled nodes.
(Formerly M3161)
17
1, 3, 54, 3834, 1027080, 1067308488, 4390480193904, 72022346388181584, 4721717643249254751360, 1237892809110149882059440768, 1298060596773261804821355107253504, 5444502293680983802677246555274553481984, 91343781554246596956424128384394531707099632640 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

REFERENCES

R. W. Robinson, Counting labeled acyclic digraphs, pp. 239-273 of F. Harary, editor, New Directions in the Theory of Graphs. Academic Press, NY, 1973.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

LINKS

T. D. Noe, Table of n, a(n) for n=1..30

A. Iványi, Leader election in synchronous networks, Acta Univ. Sapientiae, Mathematica, 5, 2 (2013) 54-82.

FORMULA

a(n) = A062738(n)/2^n, since binary relations = digraphs with loops. - Ralf Stephan and Vladeta Jovovic, Mar 24 2004

E.g.f.: log(sum n>=0, 2^(n^2-n)*x^n/n!).

a(n) = A053763(n) - (1/n) * Sum_{k=1..n-1} k*C(n,k)*a(k)*A053763(n-k). - Geoffrey Critzer, Oct 24 2012

MAPLE

b:= n-> 2^(n^2-n):

a:= proc(n) option remember; local k; `if`(n=0, 1,

      b(n)- add(k*binomial(n, k) *b(n-k)*a(k), k=1..n-1)/n)

    end:

seq(a(n), n=1..20);  # Alois P. Heinz, Oct 21 2012

MATHEMATICA

Range[0, 20]! CoefficientList[Series[D[1 + Log[Sum[2^(n^2 - n) x^n/n!, {n, 0, 20}]], x], {x, 0, 20}], x]

c[n_]:=2^(n(n-1))-Sum[k Binomial[n, k]c[k] 2^((n-k)(n-k-1)), {k, 1, n-1}]/n; c[0]=1; Table[c[i], {i, 0, 20}]  (* Geoffrey Critzer, Oct 24 2012 *)

PROG

(PARI) v=Vec(log(sum(n=0, default(seriesprecision), 2^(n^2-n)*x^n/n!))); for(i=1, #v, v[i]*=(i-1)!); v \\ Charles R Greathouse IV, Feb 14 2011

(Sage)

b = lambda n: 2^(n^2-n)

@cached_function

def A003027(n):

    return b(n) - sum(k*binomial(n, k)*b(n-k)*A003027(k) for k in (1..n-1)) / n

[A003027(n) for n in (1..13)] # Peter Luschny, Jan 18 2016

CROSSREFS

The unlabeled case is A003085.

Row sums of A062735.

Cf. A053763 (not necessarily connected), A003030 (strongly connected).

Sequence in context: A171213 A006025 A049414 * A054545 A158103 A174579

Adjacent sequences:  A003024 A003025 A003026 * A003028 A003029 A003030

KEYWORD

nonn,easy,nice

AUTHOR

N. J. A. Sloane

EXTENSIONS

Corrected and extended by Vladeta Jovovic, Goran Kilibarda

STATUS

approved

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Last modified August 16 02:15 EDT 2022. Contains 356150 sequences. (Running on oeis4.)