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%I #32 Aug 15 2019 00:39:11
%S 1,1,4,38,727,26013,1597690,149248656,18919743219,3005354096360,
%T 569226803220234,124594074249852576,30861014504270954737,
%U 8520443838646833231236,2592150684565935977152860,861079753184429687852978432,310008316267496041749182487881
%N Number of connected planar graphs on n labeled nodes.
%D Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, p. 419.
%H Gheorghe Coserea, <a href="/A096332/b096332.txt">Table of n, a(n) for n = 1..126</a>
%H M. Bodirsky, C. Groepl and M. Kang, <a href="http://dx.doi.org/10.1007/3-540-45061-0_84">Generating Labeled Planar Graphs Uniformly At Random</a>, ICALP03 Eindhoven, LNCS 2719, Springer Verlag (2003), 1095 - 1107.
%H M. Bodirsky, C. Groepl and M. Kang, <a href="http://dx.doi.org/10.1016/j.tcs.2007.02.045">Generating Labeled Planar Graphs Uniformly At Random</a>, Theoretical Computer Science, Volume 379, Issue 3, 15 June 2007, Pages 377-386.
%H O. Gimenez and M. Noy, <a href="http://arXiv.org/abs/math.CO/0501269">Asymptotic enumeration and limit laws of planar graphs</a>, arXiv:math/0501269 [math.CO], 2005.
%F This is generated by log(1+g(x)), where g(x) is the e.g.f. for labeled planar graphs, which may be computed from recurrences in Bodirsky et al. - _Keith Briggs_, Feb 04 2005
%F a(n) ~ c * n^(-7/2) * gamma^n * n!, where c = 0.00000410436110025...(A266392) and gamma = 27.2268777685...(A266390) (see Gimenez and Noy). - _Gheorghe Coserea_, Feb 24 2016
%e There are 4 connected labeled planar graphs on 3 nodes:
%e 1-2-3,
%e 1-3-2,
%e 2-1-3 and
%e 1-2
%e |/
%e 3
%o (PARI)
%o Q(n,k) = { \\ c-nets with n-edges, k-vertices
%o if (k < 2+(n+2)\3 || k > 2*n\3, return(0));
%o sum(i=2, k, sum(j=k, n, (-1)^((i+j+1-k)%2)*binomial(i+j-k,i)*i*(i-1)/2*
%o (binomial(2*n-2*k+2,k-i)*binomial(2*k-2, n-j) -
%o 4*binomial(2*n-2*k+1, k-i-1)*binomial(2*k-3, n-j-1))));
%o };
%o A100960_ser(N) = {
%o my(x='x+O('x^(3*N+1)), t='t+O('t^(N+4)),
%o q=t*x*Ser(vector(3*N+1, n, Polrev(vector(min(N+3, 2*n\3), k, Q(n,k)),'t))),
%o d=serreverse((1+x)/exp(q/(2*t^2*x) + t*x^2/(1+t*x))-1),
%o g2=intformal(t^2/2*((1+d)/(1+x)-1)));
%o serlaplace(Ser(vector(N, n, subst(polcoeff(g2, n,'t),'x,'t)))*'x);
%o };
%o A096331_seq(N) = Vec(subst(A100960_ser(N+2),'t,1));
%o A096332_seq(N) = {
%o my(x='x+O('x^(N+3)), b=x^2/2+serconvol(Ser(A096331_seq(N))*x^3, exp(x)));
%o Vec(serlaplace(intformal(serreverse(x/exp(b'))/x)));
%o };
%o A096332_seq(15) \\ _Gheorghe Coserea_, Aug 10 2017
%Y Cf. A066537, A096331, A266390, A266392, A267411.
%K nonn,hard
%O 1,3
%A _Steven Finch_, Aug 02 2004
%E More terms from _Keith Briggs_, Feb 04 2005
%E More terms from _Alois P. Heinz_, Dec 30 2015
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