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A002031 Number of labeled connected digraphs where every node has indegree 0 or outdegree 0 and no isolated nodes.
(Formerly M1707 N0676)
11
2, 6, 38, 390, 6062, 134526, 4172198, 178449270, 10508108222, 853219059726, 95965963939958, 15015789392011590, 3282145108526132942, 1005193051984479922206, 432437051675617901246918, 261774334771663762228012950, 223306437526333657726283273822 (list; graph; refs; listen; history; text; internal format)
OFFSET

2,1

COMMENTS

Also number of labeled connected graphs with 2-colored nodes with no isolated nodes where black nodes are only connected to white nodes and vice versa.

REFERENCES

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

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

LINKS

Alois P. Heinz, Table of n, a(n) for n = 2..100

Read, R. C.; Wright, E. M., Colored graphs: A correction and extension, Canad. J. Math. 22 1970 594-596.

N. J. A. Sloane, Transforms

FORMULA

Logarithmic transform of A052332.

E.g.f.: ln(Sum(exp((2^n-2)*x)*x^n/n!, n=0..infinity)). - Vladeta Jovovic, May 28 2004

a(n) = f(n,2) using functions defined in A002032. - Sean A. Irvine, May 29 2013

MAPLE

logtr:= proc(p) local b; b:=proc(n) option remember; local k; if n=0 then 1 else p(n)- add(k *binomial(n, k) *p(n-k) *b(k), k=1..n-1)/n fi end end: digr:= n-> add(binomial(n, k) *(2^k-2)^(n-k), k=0..n): a:= logtr(digr): seq(a(n), n=2..25);  # Alois P. Heinz, Sep 14 2008

MATHEMATICA

max = 18; f[x_] := Sum[ a[n]*(x^n/n!), {n, 0, max}]; coes = CoefficientList[ Series[ f[x] - Log[ Sum[ Exp[ (2^n-2)*x]*(x^n/n!), {n, 0, max}]], {x, 0, max}], x]; Table[a[n], {n, 2, 18}] /. First[ Solve[ Thread[ coes == 0]]] (* Jean-Fran├žois Alcover, Nov 08 2011, after Vladeta Jovovic *)

CROSSREFS

Cf. A001831, A001832, A002032, A047863, A052332. Essentially the same as A002027.

Sequence in context: A067106 A032111 A013703 * A184731 A005738 A055704

Adjacent sequences:  A002028 A002029 A002030 * A002032 A002033 A002034

KEYWORD

nonn,nice

AUTHOR

N. J. A. Sloane.

EXTENSIONS

More terms, formula and new title from Christian G. Bower, Dec 15 1999.

Corrected by Vladeta Jovovic, Apr 12 2003

STATUS

approved

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Last modified April 16 21:07 EDT 2014. Contains 240627 sequences.