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A002031
Number of labeled connected digraphs on n nodes where every node has indegree 0 or outdegree 0 and no isolated nodes.
16
2, 6, 38, 390, 6062, 134526, 4172198, 178449270, 10508108222, 853219059726, 95965963939958, 15015789392011590, 3282145108526132942, 1005193051984479922206, 432437051675617901246918, 261774334771663762228012950, 223306437526333657726283273822
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.
In- or outdegree zero implies loops are not admitted. Multi-arcs are not admitted. - R. J. Mathar, Nov 18 2023
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
R. C. Read, E. M. Wright, 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.: 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
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
terms = 17; s = Log[Sum[Exp[(2^n - 2)*x]*(x^n/n!), {n, 0, terms+2}]] + O[x]^(terms+2); Drop[CoefficientList[s, x]*Range[0, terms+1]!, 2] (* Jean-François Alcover, Nov 08 2011, after Vladeta Jovovic, updated Jan 12 2018 *)
CROSSREFS
Cf. A001831, A001832, A002032, A047863, A052332, A007776 (unlabeled case). Essentially the same as A002027.
Sequence in context: A067106 A032111 A013703 * A184731 A005738 A055704
KEYWORD
nonn,nice
EXTENSIONS
More terms, formula and new title from Christian G. Bower, Dec 15 1999
Corrected by Vladeta Jovovic, Apr 12 2003
STATUS
approved