%I M1707 N0676 #52 Nov 18 2023 14:17:39

%S 2,6,38,390,6062,134526,4172198,178449270,10508108222,853219059726,

%T 95965963939958,15015789392011590,3282145108526132942,

%U 1005193051984479922206,432437051675617901246918,261774334771663762228012950,223306437526333657726283273822

%N Number of labeled connected digraphs on n nodes where every node has indegree 0 or outdegree 0 and no isolated nodes.

%C 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.

%C In- or outdegree zero implies loops are not admitted. Multi-arcs are not admitted. - _R. J. Mathar_, Nov 18 2023

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

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

%H Alois P. Heinz, <a href="/A002031/b002031.txt">Table of n, a(n) for n = 2..100</a>

%H R. C. Read, E. M. Wright, <a href="http://dx.doi.org/10.4153/CJM-1970-066-1">Colored graphs: A correction and extension</a>, Canad. J. Math. 22 1970 594-596.

%H N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>

%F Logarithmic transform of A052332.

%F E.g.f.: log(Sum(exp((2^n-2)*x)*x^n/n!, n=0..infinity)). - _Vladeta Jovovic_, May 28 2004

%F a(n) = f(n,2) using functions defined in A002032. - _Sean A. Irvine_, May 29 2013

%p 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

%t 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 *)

%Y Cf. A001831, A001832, A002032, A047863, A052332, A007776 (unlabeled case). Essentially the same as A002027.

%K nonn,nice

%O 2,1

%A _N. J. A. Sloane_

%E More terms, formula and new title from _Christian G. Bower_, Dec 15 1999

%E Corrected by _Vladeta Jovovic_, Apr 12 2003