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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)
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)



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.


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).


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

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

N. J. A. Sloane, Transforms


Logarithmic transform of A052332.

E.g.f.: log(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


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


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-François Alcover, Nov 08 2011, after Vladeta Jovovic *)


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




N. J. A. Sloane


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

Corrected by Vladeta Jovovic, Apr 12 2003



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Last modified August 19 21:13 EDT 2017. Contains 290821 sequences.