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Number of labeled connected bi-point-determining graphs with n vertices (see A129583).
5

%I #21 May 07 2021 15:45:09

%S 1,1,0,0,12,252,12312,1061304,170176656,51134075424,29204599254624,

%T 32130964585236096,68873851786953047040,290164895151435531345024,

%U 2417786648013402212500060416,40014055814155246577685250570752,1318911434129029730677931158374449664

%N Number of labeled connected bi-point-determining graphs with n vertices (see A129583).

%C The calculation of connected bi-point-determining graphs is carried out by examining the connected components of bi-point-determining graphs. For more details, see reference.

%H Andrew Howroyd, <a href="/A129585/b129585.txt">Table of n, a(n) for n = 0..50</a>

%H Ira M. Gessel and Ji Li, <a href="https://doi.org/10.1016/j.jcta.2010.03.009">Enumeration of point-determining Graphs</a>, Journal of Combinatorial Theory, Series A 118 (2011) 591-612.

%F E.g.f.: 1 + x + log((Sum_{n>=0} 2^binomial(n,2)*(2*log(1+x)-x)^n/n!)/(1+x)). - Goran Kilibarda, _Vladeta Jovovic_, May 09 2007

%F E.g.f.: 1 + x + log(B(x)/(1+x)) where B(x) is the e.g.f. of A129583. - _Andrew Howroyd_, May 06 2021

%t max = 15; f[x_] := x + Log[ Sum[ 2^Binomial[n, 2]*((2*Log[1 + x] - x)^n/n!), {n, 0, max}]/(1 + x)]; A129585 = Drop[ CoefficientList[ Series[ f[x], {x, 0, max}], x]*Range[0, max]!, 1](* _Jean-François Alcover_, Jan 13 2012, after e.g.f. *)

%o (PARI) seq(n)={my(g=sum(k=0, n, 2^binomial(k,2)*x^k/k!) + O(x*x^n)); Vec(serlaplace(1+x+log(subst(g, x, 2*log(1+x+O(x*x^n))-x)/(1+x))))} \\ _Andrew Howroyd_, May 06 2021

%Y Cf. graphs: labeled A006125, unlabeled A000568; connected graphs: labeled A001187, unlabeled A001349; point-determining graphs: labeled A006024, unlabeled A004110; connected point-determining graphs: labeled A092430, unlabeled A004108; connected co-point-determining graphs: labeled A079306, unlabeled A004108; bi-point-determining graphs: labeled A129583, unlabeled A129584; connected bi-point-determining graphs: labeled A129585, unlabeled A129586; phylogenetic trees: labeled A000311, unlabeled A000669.

%K nice,nonn

%O 0,5

%A Ji Li (vieplivee(AT)hotmail.com), May 07 2007

%E More terms from Goran Kilibarda, _Vladeta Jovovic_, May 09 2007

%E a(0)=1 prepended and terms a(16) and beyond from _Andrew Howroyd_, May 06 2021