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%I A232699 #29 Sep 13 2018 13:20:37
%S A232699 1,1,1,3,15,135,1875,38745,1168545,50017905,3029330745,257116925835,
%T A232699 30546104308335,5065906139629335,1172940061645387035,
%U A232699 379092680506164049425,171204492289446788997825,108139946568584292606269025,95671942593719946611454522225
%N A232699 Number of labeled point-determining bipartite graphs on n vertices.
%C A232699 A graph is point-determining if no two vertices have the same set of neighbors. This kind of graph is also called a mating graph.
%C A232699 a(n) is always odd.
%C A232699 For every prime p > 2, a(n) is divisible by p for all n >= p.  It follows that, if m is odd and squarefree with largest prime factor q, then a(n) is divisible by m for all n >= q.  A similar property appears to hold for odd prime powers, in which case it would hold for all odd numbers.
%H A232699 Andrew Howroyd, <a href="/A232699/b232699.txt">Table of n, a(n) for n = 0..100</a> (terms 0..20 from Justin M. Troyka)
%H A232699 Ira Gessel and Ji Li, <a href="http://arxiv.org/abs/0705.0042">Enumeration of point-determining graphs</a>, arXiv:0705.0042 [math.CO], 2007-2009.
%H A232699 Andy Hardt, Pete McNeely, Tung Phan, and Justin M. Troyka, <a href="http://arxiv.org/abs/1312.0542">Combinatorial species and graph enumeration</a>, arXiv:1312.0542 [math.CO], 2013.
%F A232699 From _Andrew Howroyd_, Sep 09 2018: (Start)
%F A232699 a(n) = Sum_{k=0..n} Stirling1(n,k)*A047864(k).
%F A232699 E.g.f: sqrt(Sum_{k=0..n} exp(2^k*log(1+x))*log(1+x)^k/k!). (End)
%e A232699 Consider n = 3. The triangle graph is point-determining, but it is not bipartite, so it is not counted in a(3). The graph 1--2--3 is bipartite, but it is not point-determining (the vertices on the two ends have the same neighborhood), so it is also not counted in a(3). The only graph counted in a(3) is the graph *--*  * (with three possible labelings). - _Justin M. Troyka_, Nov 27 2013
%t A232699 terms = 20;
%t A232699 CoefficientList[Sqrt[Sum[((1+x)^2^k Log[1+x]^k)/k!, {k, 0, terms}]] + O[x]^terms, x] Range[0, terms-1]! (* _Jean-François Alcover_, Sep 13 2018, after _Andrew Howroyd_ *)
%o A232699 (PARI) seq(n)={my(A=log(1+x+O(x*x^n))); Vec(serlaplace(sqrt(sum(k=0, n, exp(2^k*A)*A^k/k!))))} \\ _Andrew Howroyd_, Sep 09 2018
%Y A232699 Cf. A006024, A004110 (labeled and unlabeled point-determining graphs).
%Y A232699 Cf. A092430, A004108 (labeled and unlabeled connected point-determining graphs).
%Y A232699 Cf. A218090 (unlabeled point-determining bipartite graphs).
%Y A232699 Cf. A232700, A088974 (labeled and unlabeled connected point-determining bipartite graphs).
%K A232699 nonn
%O A232699 0,4
%A A232699 _Justin M. Troyka_, Nov 27 2013

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